The Action of Young Subgroups on the Partition Complex

We study the restrictions, the strict fixed points, and the strict quotients of the partition complex $|\Pi_n|$, which is the $\Sigma_n$-space attached to the poset of proper nontrivial partitions of the set $\{1,\ldots,n\}$. We express the space of fixed points $|\Pi_n|^G$ in terms of subgroup posets for general $G\subset \Sigma_n$ and prove a formula for the restriction of $|\Pi_n|$ to Young subgroups $\Sigma_{n_1}\times \dots\times \Sigma_{n_k}$. Both results follow by applying a general method, proven with discrete Morse theory, for producing equivariant branching rules on lattices with group actions. We uncover surprising links between strict Young quotients of $|\Pi_n|$, commutative monoid spaces, and the cotangent fibre in derived algebraic geometry. These connections allow us to construct a cofibre sequence relating various strict quotients $|\Pi_n|^\diamond\wedge_{\Sigma_n} (S^\ell)^{\wedge n}$ and give a combinatorial proof of a splitting in derived algebraic geometry. Combining all our results, we decompose strict Young quotients of $|\Pi_n|$ in terms of"atoms"$|\Pi_d|^\diamond\wedge_{\Sigma_d} (S^\ell)^{\wedge d}$ for $\ell$ odd and compute their homology. We thereby also generalise Goerss' computation of the algebraic Andr\'e-Quillen homology of trivial square-zero extensions from $\mathbb{F}_2$ to $\mathbb{F}_p$ for $p$ an odd prime.


Exposition
Let Π n denote the poset of proper nontrivial partitions of {1, . . . , n} and write |Π n | for its geometric realisation, the partition complex. There is a well-known equivalence |Π n | (n−1)! S n−3 . However, this equivalence does not preserve the natural action of the symmetric group Σ n .
The equivariant topology of the space |Π n | arises in numerous contexts of interest: its homology group H n−3 (|Π n |, Z) is closely related to the Lie representation, its desuspended Spanier-Whitehead dual parametrises spectral Lie algebras and Goodwillie's Taylor expansion of spaces, and, as we shall prove, the homology of its strict quotients computes the cotangent fibre of certain simplicial commutative rings. Moreover, |Π n | is equivalent to the complex of non-connected graphs on n vertices, which appears in Vassiliev's work on knot theory.
In this article, we describe general fixed points, Young restrictions, and strict Young quotients of |Π n |.
Our results for fixed points and restrictions take the form of branching rules, i.e. equivariant equivalences to wedge sums of simpler spaces. We produce these rules by applying a new general algorithm which takes a G-lattice P and a list of functions (F 1 , . . . , F k ) as input and gives a rule for how to equivariantly collapse a subspace of |P − {0,1}| as output. In the simplest instance, we recover a common generalisation of work of Björner-Walker [BW83], Welker [Wel90], and Kozlov [Koz98].
Example. If we feed our algorithm Π 4 with its Σ 2 × Σ 2 -action and a suitably chosen pair of functions, it collapses the contractible subcomplex drawn with thin lines on the left and gives rise to the bouquet of circles on the right: In general, our branching rule for restrictions of Π n to Young subgroups Σ n1 × . .  Here B(m 1 , . . . , m k ) denotes the set of words in letters c 1 , . . . , c k which are lexicographically minimal among their own cyclic rotations and involve the letter c i exactly m i times. Its size is given by a famous formula due to Witt [Wit37] as The branching rule allows us to split strict Young quotients of |Π n |, and we decompose them even further by constructing a new cofibre sequence of strict orbit spaces for each d and each n even: This sequence is a "strict" analogue of the Takayasu cofibration sequence (cf. [Tak99], [Kuh01]).
Based on these topological results, we compute the homology of strict Young quotients of |Π n | with coefficients in Q and F p for all primes p. From our computations, we can read off the algebraic André-Quillen homology of trivial square-zero extensions over these fields, thereby generalising work of Goerss over F 2 and establishing a key tool in the emerging field of deformation theory in characteristic p.

Statement of Results
We shall begin by describing our general combinatorial branching algorithm and then proceed to explain its consequences for fixed points and restrictions of partition complexes and Bruhat-Tits buildings. Finally, we will discuss several computations and conceptual connections concerning strict quotients of partition complexes.
Complementary Collapse. Fix a finite group G acting in an order-preserving manner on a finite lattice P, by which we mean a poset containing all binary meets x ∧ y and joins x ∨ y. We let F P denote the collection of nondegenerate chains in P and write P = P − {0,1} for the poset obtained by removing the minimum0 and the maximum1 from P. In order to describe our algorithm, we introduce the following notion: Definition 4. 16. A function F : F P → P is called an orthogonality function if (1) F is G-equivariant and increasing (i.e. y ≤ F (σ) for every σ ∈ F P and every y ∈ σ.) (2) For any σ = [y 0 < · · · < y m ] ∈ F P and z > y m , the following subposet is discrete: Lists F = (F 1 , . . . , F n ) of orthogonality functions are examples of "orthogonality fans" (cf. Definition 4.8), and there is a notion for when a chain σ = [y 0 < · · · < y r ] is orthogonal to a fan F, written σ ⊥ F (cf. Definition 4.9). Using discrete Morse theory (cf. [For98], [Fre09]), we prove: Here P (a,b) denotes the subposet of elements z with a < z < b, and the unreduced suspension of a space X is denoted by X . If X is pointed, we write ΣX for its reduced suspension. An equivariant simple homotopy equivalence is an equivariant equivalence which can be obtained by iterated elementary expansions and collapses (cf. Definition 2.11).
Complementary collapse constitutes a powerful tool in poset topology, as we will demonstrate now.
Fixed Points. We write P n for the lattice of partitions of n = {1, . . . , n} and set Π n = P n −{0, 1}. Given a subgroup G ⊂ Σ n , it is natural to ask: Question. What is the W Σn (G) = N Σn (G)/G-equivariant simple homotopy type of |Π n | G ?
If G acts transitively on n = {1, . . . , n}, then n can be identified, as a G-set, with the coset space G/H for H the stabiliser of the element 1, say. For such transitive actions, it is not difficult to show that the poset of G-invariant partitions of G/H is isomorphic to the poset of subgroups of G that contain H. This is Lemma 6.5 (it had appeared in the paper of White-Williamson [WW76, Lemma 3], and they attributed the result to Klass [Kla73]). For general G ⊂ Σ n , this question is more difficult. Group actions on posets fall into two categories: either all orbits are equivariantly isomorphic, in which case the action is called isotypical, or they are not, in which case the action is called non-isotypical.
Complementary collapse reduces the above question to the well-understood transitive case: Theorem 6. 2. If G acts isotypically, we relabel n so that G is a transitive subgroup of a diagonally embedded Σ d Then there is a W Σn (G) = N Σn (G)/G-equivariant simple homotopy equivalence Here we used the following notation: given a subgroup H ⊂ G and a pointed H-space X, the induced G-space Ind G H (X) = G + ∧ H X is the wedge of |G/H| copies of X with its natural G-action. In the remaining case, we have: Lemma 6. 3. If G acts non-isotypically, then |Π n | G is W Σn (G)-equivariantly collapsible.
Remark. This lemma also has a very straightforward and direct proof. It has been observed independently by Markus Hausmann.
In [ADL16], the authors proved that the only p-groups P for which the fixed point space |Π n | P is not contractible are elementary abelian groups acting freely on n. It was also observed that if n = p k and P ∼ = F k p acts freely and transitively on n, then Π P n is isomorphic to the poset BT(F k p ) of proper non-trivial subgroups of P , which is closely related to the Tits building for GL k (F p ). Combining these observations with our Theorem 6.2 and Lemma 6.3, we can complete the calculation and describe all fixed point spaces of the partition complex with respect to p-groups: Corollary 6.8. Let P ⊂ Σ n be a p-group.
(1) If P ∼ = F k p is elementary abelian acting freely on n with n = mp k (where p may divide m), we write Aff F k p = N Σ p k (F k p ) = F k p GL k (F p ) for the affine group associated with F k p and define Aff F k p Σm := N Σn (F k p ) = (F k p ) m (GL k (F p ) × Σ m ). Then there is an Aff F k p Σm -equivariant simple homotopy equivalence As before, BT(F k p ) denotes the poset of proper nontrivial subspaces of the vector space F k p . Nonequivariantly, this implies that |Π n | P is a bouquet of (m − 1)! · p k(m−1)+( k 2 ) spheres of dimension m + k − 3.
(2) If the action of P is not of this form, then |Π n | P is W Σn (P )-equivariantly contractible.
Our techniques can also be used to examine fixed points under iterated Wreath products: Lemma 6. 6. Suppose that d | n and that d factors as a product d = d 1 · · · d l of integers d 1 , . . . , d l > 1 with l > 1. Consider the iterated wreath product Σ d1 · · · Σ d l as a subgroup of Σ n via the diagonal embedding. Then the space |Π n | Σ d 1 ··· Σ d l is collapsible. Restrictions. We proceed to our results concerning the restrictions of |Π n | to Young subgroups. Fix positive integers n 1 , . . . , n k with n = n 1 +. . .+n k . As before, we write B(n 1 , . . . , n k ) for the set of words in letters c 1 , . . . , c k which are lexicographically minimal among their rotations and involve the letter c i precisely n i times. Words with this minimality property are called Lyndon words.
The asserted equivalence is pointed, where the basepoint of |Π n | is given by the above partition x.
If d > 0 divides all of n 1 , . . . , n k , then Σ n d d is a subgroup of Σ n1 × · · · × Σ n k . We obtain a diagonal inclusion Σ d → Σ n1 × · · · × Σ n k and will identify Σ d with its image under this map.
By Σ −1 (S n d −1 ) ∧d , we mean a desuspension of the sphere (S n d −1 ) ∧d with its natural Σ d -action. Such a desuspension exists since there is a Σ d -equivariant homeomorphism between S d and the suspension of the reduced standard representation sphere of Σ d . We should remark that our definition of Π d only works well for d ≥ 2. For d = 2, |Π 2 | is the empty set, which we identify with the (−1)-dimensional sphere, since its unreduced suspension |Π 2 | is homeomorphic to S 0 . We decree that |Π 1 | is the (−2)-dimensional sphere. In practice, this means that the space |Π 1 | is undefined, but for any suspension X , we define X ∧ |Π 1 | := X.
Our Theorem 5.10 is a strengthening of the main result of [AK98]. In [op. cit], the authors applied Goodwillie calculus to the Hilton-Milnor theorem, which is a decomposition result for the topological free group generated by a wedge sum of connected spaces. It was possible to prove a weak version of Theorem 5.10 which holds only after one (a) makes the group action pointed-free, (b) applies the suspension spectrum functor. Moreover, the map defining this equivalence was not constructed explicitly in [op. cit.].
In this work, we first give an explicit point-set level description of the map in (1.1) as a collapse map and then prove that it is an equivariant simple homotopy equivalence of spaces. Such an explicit description is desirable for applications.
For example, partition complexes show up in the study of spectral Lie algebras, i.e. algebras over the spectral Lie operad Lie defined by Salvatore [Sal98] and Ching [Chi05]. The n th term of Lie is given by Map Sp (S 1 , (S 1 ) ∧n ) ∧ D(Σ|Π n | ), where D denotes Spanier-Whitehead duality. Its structure maps are constructed by "grafting" weighted trees. The homotopy and homology groups of spectral Lie algebras form graded Lie algebras.
In Section 3.8 and 3.9, we link the simplicial collapse maps in Theorem 5.10 to the structure maps of the spectral Lie operad. This allows us to give a concrete description of free spectral Lie algebras on many generators. Indeed, suppose that we are given spectra X 1 , . . . , X k . Let w be a Lie word in letters c 1 , . . . , c k involving the letter c i precisely |w| i times. We can then define a natural map F w : X . . ∨ X k ) to the free spectral Lie algebra on X 1 ∨ . . . ∨ X k . The induced map π * (X 1 ) ⊗|w|1 ⊗ . . . ⊗ π * (X k ) ⊗|w| k − → π * (Free Lie (X 1 ∨ . . . ∨ X k )) has the effect of sending an element (x 1 1 ⊗ . . . . .) to the element obtained by first replacing, for all i, the occurrences of c i in w by x 1 i , . . . , x |w|i i from left to right, and then evaluating this Lie product in the Lie algebra π * (Free Lie (X 1 ∨ . . . ∨ X k )). Theorem 5.10 implies: Corollary 5. 13. Summing up the induced maps over all words w lying in the Lyndon basis B k for the free Lie algebra on k letters, we obtain an equivalence of spectra w∈B k F w : We can also give an asymmetric decomposition for Young restrictions of |Π n |. For this, we fix a Young subgroup Σ A × Σ B1 × · · · × Σ B k ⊂ Σ n . Complementary collapse implies: Theorem 5.15 (Symmetry Breaking). There is a simple Σ A × Σ B1 × · · · × Σ B k -equivariant homotopy equivalence spaces |Π n | −→ A=A1 ··· Ar, Ai =∅ fi:Ai →B s.t. im(fi+1)⊂im(fi) Σ −1 S |A1| ∧ · · · ∧ S |Ar| ∧ |Π B | .
Partition complexes can be thought of as Bruhat-Tits buildings over "the field with one element".
In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary collapse then has a neat analogous consequence for parabolic restrictions of Bruhat-Tits buildings.
Let V be a finite-dimensional vector space over a finite field k. Fix a flag A = [A 0 < · · · < A r ] of proper nonzero subspaces of V , and write P A for the associated parabolic subgroup. Choose a complementary flag B = [B 0 < · · · < B r ] to A with corresponding parabolic subgroup P B . Write L AB = P A ∩ P B for the intersecting Levi. Complementary collapse shows: Lemma 5. 17. There is a P A -equivariant simple equivalence | BT(gr i (B))| ).
Here gr i (B) = B i /B i−1 for i = 1, . . . , r and we set gr 0 (B) = B 0 and gr r+1 (B) = V /B r . We are therefore reduced to answering the above questions for spaces of the form Σ −1 (S ) ∧d ∧ Σ d |Π d | .
Fix a prime p. In order to compute the F p -homology of the spaces in question, we consider the graded Mackey functor with µ * (T ) = H * (S n ∧ Σn T + , F p ). There is a spectral sequence of signature H Br s (|Π n | ; µ t ) ⇒ H s+t (S n ∧ Σn |Π n | , F p ) from the Bredon homology to the desired homology of strict quotients.
If is odd, then this Mackey functor µ * has desirable properties and indeed satisfies the conditions of the main Theorem 1.1. of [ADL16]. Using this, we compute in our final Section 9: Theorem 9.1. Let ≥ 1 be an integer, assumed to be odd whenever the prime p is odd. If n is not a power of p, then H * (|Π n | ∧ Σn S n , F p ) is trivial.
If n = p a , then H * (|Π p a | ∧ Σ p a S p a , F p ) has a basis consisting of sequences (i 1 , . . . , i a ), where i 1 , . . . , i a are positive integers satisfying: (1) Each i j is congruent to 0 or 1 modulo 2(p − 1).
In order to reduce the case where is even to the case where is odd, a new conceptual insight is required.
The key observation to be made is that for p = 2, the above basis has the same form as the algebraic André-Quillen homology AQ F2 * (F 2 ⊕ Σ F 2 ) of the trivial square zero extension of F 2 by a generator in simplicial degree j as computed by Goerss (cf. [Goe90]). This similarity suggests that strict quotients of the partition complex are also the André-Quillen homology of suitably defined objects.
Indeed, the correct structure to consider here is that of a (strictly) commutative monoid space, i.e. a space X together with a continuous, commutative, and associative multiplication law, a unit 1, and a contracting element 0. Heuristically, these monoids give "simplicial commutative rings over F 1 ".
More formally, we consider the pointed model category CMon aug of strictly commutative monoid spaces augmented over S 0 , the monoid with two distinct elements 0 and 1. Any pointed space X gives rise to an augmented commutative monoid space S 0 ∨ X by declaring that a · b = 0 unless a = 1 or b = 1. We call this the trivial square zero extension of S 0 by X. There is a natural notion of André-Quillen homology, denoted by AQ, for these commutative monoid spaces.
The following result establishes the crucial connection to strict orbits of the partition complex: We achieve the aforementioned reduction "from even to odd ' by constructing a new cofibre sequence. We shall outline its construction. There is a natural notion of suspension Σ ⊗ in the pointed model category CMon aug .
For every pointed space, we produce a natural "EHP-like" sequence of maps The first map in this sequence is subtle to define and requires us to work with point-set models for strictly commutative monoid spaces. We then prove: Theorem 7. 44. If S n is a sphere of even dimension n ≥ 2, then the EHP-sequence is in fact a homotopy cofibre sequence of strictly commutative monoid spaces.
Here we use the convention that the space on the left is contractible if d is odd.
But first, we give a conceptual interpretation of the homology of strict Young quotients of |Π n |, thus explaining the aforementioned similarity between the F 2 -homology of |Π d | ∧ Σ d (S ) ∧d with odd and the algebraic André-Quillen homology of the trivial square-zero extensions of F 2 as computed by Goerss. Given any ring R, we can apply the reduced chains functorC • (−, R) to a commutative monoid space X and obtain a simplicial commutative R-algebra. Heuristically, we extend scalars from F 1 to R. This operation intertwines André-Quillen chains for strictly commutative monoid spaces with the classical algebraic André-Quillen chains for simplicial commutative R-algebras (cf. Lemma 7.34). Moreover, the functorC • (−, R) sends the trivial square-zero extension S 0 ∨ X to the trivial square zero extension R ⊕C • (X, R). When combined with Lemma 8.4, these observations show that the singular homology of strict Young quotients of the doubly suspended partition complex Σ|Π n | computes algebraic André-Quillen homology: Theorem 8. 8. If X is a well-pointed space and R is a ring, then The algebraic André-Quillen cohomology groups of trivial square-zero extensions of R are of particular interest since they parametrise the operations which act naturally on the André-Quillen cohomology of any simplicial commutative R-algebra.
Theorem 8.8 allows us to observe that our splitting in Corollary 8.2 and our cofibration sequence in Theorem 8.5 imply corresponding results for the algebraic André-Quillen homology over any ring. Over F 2 , these were first proven by Goerss [Goe90]. We find it remarkable that our combinatorial methods have these consequences in derived algebraic geometry.
For R = F p , the above homology group has a basis indexed by sequences (i 1 , . . . , i a , e, w), where w ∈ B k (n 1 , . . . , n k ) is a Lyndon word and e lies in {0, }. Here = 1 if p is odd and |w| is even or if |w| = 0. Otherwise, we set = 0. The sequence i 1 , . . . , i a consists of positive integers satisfying: (1) Each i j is congruent to 0 or 1 modulo 2(p − 1).
A sequence (i 1 , . . . , i a , e, w) satisfying the conditions in the theorem above is called allowable with respect to R = Q or R = F p , respectively. We can read off the answer to the question raised above: Corollary 8. 15. Let n = n 1 + . . . + n k .
The vector space H * (|Π n |/ Σn 1 ×...×Σn k , Q) has a basis consisting of sequences (e, w ∈ B(m 1 , . . . , m k )) which are allowable with respect to Q and satisfy m i (1 + e) = n i for all i.
We have seen in Corollary 8.2 that strict Young quotients of partition complexes split as wedge sums of quotient spaces of the form Σ −1 (S ) ∧d ∧ Σ d |Π d | (which in turn sit in the cofibre sequences established in Theorem 8.5).
We study the homotopy type of these spaces and prove that Σ −1 (S ) ∧d ∧ Σ d |Π d | is: (1) rationally contractible unless d = 1 or d = 2 and even (this is a special case of Theorem 8.14).
Historical Review. This paper subsumes the preprint [Aro15], a part of the second author's thesis [Bra], and additional joint work. We give a brief historical review. The short discrete Morse-theoretic proof of Theorem 6.2 concerning the fixed points of partition complexes was found by the second-named author in the spring of 2014 and presented at a short presentation in Bonn in May 2015. In late August of the same year, the first-named author made his preprint publicly available. Here, the first-named author follows a very different path: He first establishes Theorem 6.2 for certain wreath products. Using this, he proceeds to prove Theorem 5.10 by reducing it to the algebraic Hilton-Milnor theorem on the level of cohomology. From this, he then deduces Theorem 6.2 for all subgroups. Strictly speaking, only the non-simple versions of these two statements were covered. The preprint also contains several other statements, in particular a version of Lemma 8.6 and Corollary 8. 23. The preprint motivated the second-named author to strengthen his discrete Morse theoretic methods, develop the theory of orthogonality fans, and thereby give the direct combinatorial proof of the strengthened version of Theorem 5.10 presented in this paper. In this approach, fixed points and restrictions are computed independently by applying a general combinatorial technique.
Acknowledgements. The first author's initial impetus for this work was provided by Ralf Donau's paper [Don12] and his desire to understand its connection with [AK98], and he is grateful to Donau for several useful email exchanges, and in particular for sharing the results of some of his computeraided calculations of the homology groups of quotient spaces of |Π n |.
The second author's interest in the partition complex was initially sparked by the study of the Morava E-theory of spectral Lie algebras in his thesis, and he would like to thank his advisor Jacob Lurie for his guidance and support. The second author was supported by an ERP scholarship of the German Academic Scholarship Foundation and by the Max Planck Institute for Mathematics in Bonn.
Both authors thank the Hausdorff Research Institute for Mathematics in Bonn for its hospitality.

Topological Preliminaries
We will now set up the basic framework used in the subsequent sections of this article. 2.1. Combinatorial Models for Spaces. We briefly review the links between several different models for spaces commonly used in homotopy theory and combinatorial topology. Write Top for the category of compactly generated weak Hausdorff spaces. We will simply call its objects "spaces". All limits and colimits will be implicitly computed in this category rather than the category of mere topological spaces (cf. [McC69], where compactly generated weak Hausdorff spaces are called "compact spaces").
In homotopy theory, we often use the category sSet of simplicial sets, i.e. contravariant functors from the category of nonempty linearly ordered finite sets ∆ to the category of sets. The Yoneda embedding i : ∆ → sSet sends the ordered set [n] := {0 < . . . < n} to the simplicial n-simplex.
In combinatorial topology, the following notion is commonly used: A simplicial complex is a pair (V, F ) consisting of a set V of vertices and a set F ⊂ P(V ) of finite nonempty subsets of V , called faces, such that F is closed under passing to nonempty subsets and contains all singletons. A morphism (V, F ) → (V , F ) of simplicial complexes is a map V → V sending subsets in F to subsets in F . Let sCpl be the resulting category.
Let Fin + be the category of nonempty finite sets. We can define a functor Fin + → sCpl by sending a set B to the simplicial complex (B, P(B)) modelling a simplex with B vertices. Yet another common model is given by the category CW of CW-complexes.
In order to link these models, we use the following gadget (cf. [Gra00], [Law88], [RT03]): The category SymsSet of symmetric simplicial sets is given by the category of contravariant functors from Fin + to Set.
There is a natural diagram The vertical arrows are given by Yoneda embeddings, the functor U forgets the order, the functor L is the colimit-preserving extension making the diagram commute, the functor F sends a finite set B to the simplex on B vertices, and the functor | − | extends F in a colimit-preserving way.
Every simplicial complex X gives a symmetric simplicial set B → Map sCpl ((B, P(B)), X), and this assignment is in fact fully faithful. Writing Po for the category of posets, there is a nerve functor N • : Po → sSet (defined by considering posets as categories) and an order complex functor N : Po → sCpl (defined by sending a poset P to the simplicial complex whose vertices are the elements of P and whose faces are given by all subsets which are chains in P ). These constructions are compatible and the following diagram commutes: We will abuse notation and denote all arrows landing in CW or in Top by | − |.
For the rest of this section, we fix a finite group G. We invite the reader to recall the notion of a G-CW-complex from [Lüc89] -note that this is not the same as a G-object in CW-complexes. We write Po G , sCpl G , sSet G , and SymsSet G for the categories of objects with G-action in the undecorated versions of these respective categories. One can then obtain a G-equivariant version of diagram (2.3) on the preceding page. A similar diagram exists in the pointed setting.

Basic Examples.
Many posets have an initial and/or final element which we will denote bŷ 0 and/or1. Given a poset P, we define P := P − {0,1}.
Its geometric realisation is the topological boundary ∂∆ m−1 of ∆ m−1 , i.e. the sphere S m−2 . It is well-known, and not difficult to check, that the order complexes of B − {0} and B m are isomorphic to the barycentric subdivisions of C m and of its simplicial boundary, respectively (see Example 2.4). There is a Σ m -equivariant homeomorphism |B m | S m−2 to the doubly desuspended representation sphere of the standard representation of Σ m . It is well-known and not difficult to prove that this action is homeomorphic to the action of Σ m on the unit sphere in R m−1 , where R m−1 is endowed with the reduced standard action of Σ m . 2. 3. Equivariant Homotopy Theory. Let G be a finite group. All of our spaces will be homeomorphic, as G-spaces, to geometric realisations of G-simplicial sets. A (finite) pointed G-space X can be written as a (finite) homotopy colimit of spaces G/H + , where H is an isotropy group of X.
A G-map f : X → Y is a G-equivariant weak equivalence if it induces weak equivalences of fixed point spaces f H : X H → Y H for every subgroup H of G. It is well-known that for nice spaces, an equivariant weak equivalence is an equivariant homotopy equivalence. It is in fact not necessary to check the fixed points condition for every subgroup of G -isotropy groups of X and Y suffice.
Suppose that K and H are subgroups of G and that X is a pointed space with an action of H. One may form the G-space Ind G H X := G + ∧ H X and consider the space of K-fixed points (Ind G H X) K with its action by the Weyl group W G (K) = N G (K)/K. Write N G (K; H) = {g ∈ G | g −1 Kg ⊂ H} and note that H acts on this set from the right. The following is well-known: , where the Weyl group W G (K) acts through the natural left action of N G (K) on the set N G (K; H). Hence if K is not subconjugate to H, then (Ind G H X) K ∼ = * . We will now provide some tools which make it possible to compute the effect of Theorem 5.10 on fixed points. This can be used to give an alternative proof of some of our results in Section 6.2, where we have chosen a more direct approach. Let G ⊂ Σ d ⊂ Σ n , where d | n and Σ d is embedded diagonally in Σ n (observe the change in the role of G). Lemma 2.7. Let σ ∈ N Σn (G; Σ d ) be an element of Σ n that conjugates G to a subgroup of Σ d . Then there is an element η ∈ Σ d such that conjugation by η on G is the same as conjugation by σ.
Proof. An inclusion of G into Σ n is the same thing as an effective action of G on n. If the inclusion factors through Σ d , then n is isomorphic, as a G-set, to a disjoint union of n d copies of the set d, which is equipped with an effective action of G. Specifying an element σ of Σ n that conjugates G into a subgroup of Σ d is the same as specifying a second action of G on d (let d denote d with the second action of G), a group isomorphism σ G : G → G, and a bijection σ : Let C W (G) denote the centraliser of G in W .
By Lemma 2.7 for every such w one can find an η ∈ Σ d such that conjugation by η on G coincides with conjugation by w. It follows that wη −1 ∈ C W (G). Thus every element of N W (G; Σ d )/Σ d has a representative in C W (G). Clearly two elements of C W (G) differ by an element of Σ d if and only if they differ by an element of C Σ d (G). The corollary follows.
In particular, w can always be chosen in the centraliser of G, so X w −1 Gw = X G . It follows that ( hand side is isomorphic to C W (G) + ∧ CΣ d (G) X G but the latter presentation is better in the sense that it gives the correct action of C W (G). 2.4. Simple Equivariant Homotopy Theory. We briefly review the basics of simple equivariant homotopy theory. We begin by looking at simplicial complexes and recall a notion from [Koz15]: (1) There is exactly one face in F which properly contains σ.
(2) For every g ∈ G with gσ = σ, there does not exist a simplex which contains both gσ and σ.
(3) F is obtained from F by deleting all faces which contain gσ for some g ∈ G.
There is a corresponding notion for G-CW-complexes (cf. [Lüc89]). Let D k be the k-dimensional disc.
Definition 2.11. An elementary expansion consists of a pushout of G-CW-complexes where D n−1 → D n includes the lower hemisphere D n−1 → S n of the bounding sphere S n of D n , and f (G/H × ∂D n−1 ) ⊂ X n−2 , f (G/H × D n−1 ) ⊂ X n−1 , where X k denotes the k-skeleton of X.
Given a subspace X of Y for which X → Y is an elementary expansion, we call any strong G- Geometric realisation takes simplicial to space-level join. There is a corresponding notion for posets: Definition 2.20. The join of two posets P and Q is defined to be the poset P * Q whose underlying set is given by P Q and whose partial order is defined by keeping the old order on P and Q, and declaring that every element of P is smaller than every element of Q.
It is easy to see that the order complex of P * Q is isomorphic to the simplicial join of the two order complexes. It follows that |P * Q| |P| * |Q|.

Stars and Links.
Let P be a poset. Let σ = [x 0 < · · · < x k ] be a non-empty chain in P. The star decomposes as a join St(σ) There is a similar decomposition for the link as Lk(σ) ∼ = P <x0 * P (x0,x1) * · · · * P >x k .

Indexed Wedges.
We recall the theory of indexed wedges (cf. Section 2.2. 3. of [HHR16]). Given a G-set J, we write B J G for the category with objects Ob(B J G) = J and morphisms The composition law is evident. Given a functor X : Clearly, we can rewrite this indexed wedge sum as an induction: Notation 2.25. If S is a G-stable set of nondegenerate simplices in the G-simplicial set X, let X −S be the simplicial subset of simplices which do not contain a simplex in S.
We will now remove simplices from the nerve of a G-poset P. Here, we drop N • from our notation whenever we geometrically realise, e.g. write |P −S | instead of |N • (P) −S |.
Lemma 2.26. Let S be a G-stable family of strictly increasing chains in P such that no two chains have a common refinement. The following diagram induces isomorphisms on vertical quotients: Proof. Any simplex in N d (P)−N d (P) −S contains exactly one σ ∈ S and thus has unique preimages lying in N d (St(σ)) − N d (St(σ)) −σ and N d (P) − N d (P) −σ . Proposition 2.24 proves the claim.
2. 9. Suspensions and Products. We let S 1 We will make use of this well-known simplicial model for Σ|X| in the rest of this work.
Using our models for reduced-unreduced suspensions, the following observation is straightforward (cf. Theorem 5.1. in [Wal88] for a related claim): Proposition 2.28. For i = 1, . . . , r, let P i be a poset with distinct minimal and maximal elementŝ 0 i and1 i . Then there is a natural cellular homeomorphism Proof. This follows by observing that there is a natural isomorphism of simplicial sets

Explicit Collapse.
Let P be a G-poset with distinct minimal element0 and maximal element1. Given a chain σ = [y 0 < . . . < y r ] in P = P − {0,1}, we write St(σ) for the star of σ in P. We represent points P in Such an expression represents the basepoint precisely if one of the parameters s i vanishes.
We represent points in Σ|P (0,y0) | ∧ Σ|P (y0,y1) | ∧ . . . ∧ Σ|P (yr,1) | by (r + 2)-tuples where we use the convention y −1 =0 and y r+1 =1. Such a tuple represents the basepoint if some l i −1 or l i ni+1 vanishes. We now define by sending points P represented as above to If one of the C i vanishes, we map P to the basepoint. It is not hard to check that this map is well-defined, continuous, cellular, Stab(y 0 ) ∩ · · · ∩ Stab(y r )-equivariant, and that it has a continuous inverse.
Given a G-stable family S of strictly increasing chains in P such that no two chains have a common refinement, we obtain an explicit description of the collapse maps in the simple equivalence We represent points P in the star | St(σ)|/ | St(σ) −σ | of a chain σ = [y 0 < . . . < y r ] by expressions The unsuspended collapse map then sends such a point P to the (r + 2)-tuple where we use the additional convention that s −1 = s r+1 = 0. If one of the C i vanishes, we again map P to the basepoint. We used well-known models for the two unreduced suspensions |P (0,y0) | and |P (yr,1) | . The unsuspended collapse maps again interact well with the action by G.
We will also often consider the collapse maps obtained by postcomposing the above maps to wedge summands with the "projection" to an individual summand. 2.11. Spectra. In our study of spectral Lie algebras, we shall use the closed symmetric monoidal model category (Sp, ∧, S) of S-modules, endowed with the smash product, as model for spectra (cf. [EKMM97]). We urge the reader to not get too distracted by this refined homotopical notion as many of our later arguments will merely require the homotopy category hSp of spectra.
There is an adjunction Σ ∞ : Top * Sp : Ω ∞ whose left component Σ ∞ is monoidal. Given S-modules X, Y ∈ Sp, we let Map Sp (X, Y ) be the mapping spectrum between X and Y and write D(X) = Map Sp (X, S) for the Spanier-Whitehead dual of X. Using the functor Σ ∞ , we can observe that the category Sp is both tensored and cotensored over Top * . We will suppress Σ ∞ from our notation whenever the context implies that we are working in spectra. We will now establish two results which tell us that several strict constructions are homotopically well-behaved: Assume that the map f : X → Y of spectra is (1) either obtained by applying Σ ∞ to a weak equivalence of well-pointed spaces in Top * (2) or a weak equivalence of spectra with X cofibrant and Y a suspension spectrum.
Proof. Let S c → S be a cofibrant replacement of the sphere spectrum.
For (1), Theorem 24. 3.1 of [MS06] shows that S c ∧ X → S c ∧ Y is a weak equivalence. The spectra S c ∧ X, S c ∧ Y are seen to be cofibrant by combining Theorem VII. 4.6. in [EKMM97] with Proposition 10. 3 The left horizontal arrows below are weak equivalences by Lemma 4.2.7 of [Hov99]: For claim (2), we first note that smashing with S c preserves weak equivalences by Theorem III. 3.8. of [EKMM97] and then follow a parallel argument.
Proposition 2. 30. Let X, Y ∈ Top * be well-pointed spaces. Fix weak equivalences from cofibrant spectra D(Y ) → D(Y ), X → X, Y → Y . Then, we have: is a weak equivalence of spectra whenever X is a CW-complex.

Proof.
We again let S c → S be a cofibrant replacement of S.
For (1), we consider the following square: The vertical arrows are weak equivalences by the very strong commutative monoid axiom of [Mur15], asserting that smashing S c → S with any spectrum gives a weak equivalence. Theorem III. 3.8. of [EKMM97] implies that smashing with S c preserves weak equivalences. The top horizontal arrow is a weak equivalence as it is obtained by smashing two weak equivalences between cofibrant spectra (S c ∧ X, S c ∧ Y are cofibrant by the same argument as in Proposition 2.29).
For (2), we consider the diagram The vertical arrows are equivalences since smashing with S c → S preserves weak equivalences. The top right horizontal arrow is an equivalence by Theorem III. 3.8. of [EKMM97] since S c ∧ X is a cell S-module. The top left horizontal arrow is an equivalence as it is obtained by smashing weak equivalences between cofibrant S-modules.

Lie Algebras and the Partition Complex
We will now discuss the connection between partition complexes |Π n | and the theory of classical and spectral Lie algebras. Moreover, we shall explain some straightforward algebraic consequences of our later topological results on Young restrictions of |Π n | in Section 5.1. 3.1. Free Lie Algebras. We begin by recalling the following basic notion: In fact, Lie[S] can be constructed as the quotient of the free non-associative and non-unital algebra on {c s } s∈S by the two-sided ideal generated by the antisymmetry and the Jacobi relation.
Throughout the paper, monomials are understood to be in non-commutative and non-associative variables. We will also refer to them as parenthesised monomials. Elements of Lie[S] can be represented as linear combinations of parenthesised monomials in the letters {c s } s∈S . As is customary in the context of Lie algebras, we will use square brackets to indicate the parenthesisation.  There is a second common notion of Lie algebras over the integers: Isotropic Lie algebras have the well-known property that the free isotropic Lie algebra Lie i [V ] on a free abelian group V is again a free abelian group (cf. Corollary 0.10 in [Reu03]) . This is not true for free Lie algebras in the sense of Definition 3.1. For example, the free Lie algebra Lie[c 1 ] on one generator c 1 has underlying Z-module Z ⊕ Z/2 generated by c 1 and [c 1 , c 1 ] with 2[c 1 , c 1 ] = 0.
However, totally isotropic Lie algebras are overly restrictive in our context: first, they cannot be defined as algebras over an operad. Moreover, several natural examples of interest (related to algebraic André-Quillen homology groups or spectral Lie algebras over F 2 , cf. [Goe90], [Cam16]) only satisfy the weaker antisymmetry axiom [u, v] = −[v, u]. We will therefore focus on Lie algebras.

Lyndon Words.
We will use the following classical notion (cf. [Shi58], [CFL58]): Definition 3. 4. A word w in letters c 1 , . . . , c k is a (weak) Lyndon word if it is (weakly) smaller than any of its cyclic rotations in the lexicographic order with c 1 < · · · < c k . Write B(n 1 , . . . , n k ) (or B w (n 1 , . . . , n k )) for the set of (weak) Lyndon words which involve c i precisely n i times.
A Lyndon word w of length > 1 can be written uniquely as w = uv with u < v both Lyndon words and u as long as possible -this is the (left) standard factorisation. Given any two Lyndon words u < v, the word w = uv is again a Lyndon word. The factorisation w = uv is standard if and only if v is a letter or it has standard factorisation v = xy with x ≤ u.
There is a unique injection φ from the set of Lyndon words in c 1 , . .  Any weak Lyndon word w can be written uniquely as w = u d with u a Lyndon word. We call d the period of w. This gives an identification B w (n 1 , . . . , n k ) = d| gcd(n1,...,n k ) B( n1 d , . . . , n k d ). Now let S be a finite set and assume g : S → {1, . . . , k} is a map whose fibre C i over i has size n i . Definition 3.6. An (S, g)-labelling of a weak Lyndon word w = u d in letters c 1 , . . . , c k is represented by a bijection f from S to the letters of w with the property that for each s ∈ C i , the symbol f (s) is c i . Two bijections are considered to represent the same labelling if one can be obtained from the other by permuting the various copies of u in w.
Write B w (S,g) (n 1 , . . . , n k ) for the set of (S, g)-labelled weak Lyndon words in B w (n 1 , . . . , n k ). This set comes endowed with a natural (left) Σ C1 × · · · × Σ C k -action: given a permutation h and a weak Lyndon word w whose labelling is represented by the bijection f from S to the letters of w, we form a new labelled word h · w by changing f to f • h −1 .
Labelled words can be used to define multilinear monomials containing each generator once: Definition 3.7. Let w be a Lyndon word with an (S, g)-labelling represented by the bijection f . The (S, g)-resolutionw f ∈ Lie[S] of w is represented by the multilinear monomial obtained from w by replacing each letter in w with its preimage in S under f .
We observe that this resolution procedure intertwines the action on labels in Definition 3.6 with the action on words in Section 3.
Remark 3.8. Given a decomposition n 1 + . . . + n k = n, there is a standard choice for (S, g): we take S = n = {1, . . . , n} for n = n i and use the unique order-preserving map g : n → k with |g −1 (i)| = n i for all i. We write B w n (n 1 , . . . , n k ) := B w (S,g) (n 1 , . . . , n k ). The quotient of B w (S,g) (n 1 , . . . , n k ) by the group Σ n1 × . . . × Σ n k can be identified with the set of weak Lyndon words B w (n 1 , . . . , n k ) = d| gcd(n1,...,n k ) B( n1 d , . . . , n k d ) containing the i th generator n i times. Each orbit in the Σ n1 × . . . × Σ n k -set B w n (n 1 , . . . , n k ) contains a unique labelled weak Lyndon word w = u d with the property that for each copy of u, all labels are congruent mod d and increase on all occurrences of a letter c i in this copy from left to right. We call this the standard labelling of the underlying weak Lyndon word w = u d . The various copies of u then partition n as

The Algebraic Lie Operad.
Definition 3. 9. Given a finite set S, the Lie representation Lie S of the symmetric group Σ S is given by the submodule of Lie[S] spanned by all words which contain each generator c s once.
It is well-known that Lie S is a free abelian group. The Σ S -modules Lie S assemble to the Lie operad.
Hence every bijection i∈D S i φ → S gives rise to a homomorphism s φ : Lie D ⊗ i∈D Lie Si → Lie S which is natural in the bijection φ and with respect to permutations of the sets S i .
Let us describe the map s φ explicitly. Recall that Lie S is generated by multilinear monomials in variables {c s } s∈S . Let w D ∈ Lie D and { w Si ∈ Lie Si | i ∈ D } be represented by monomials.
Then s φ (w D ⊗ i∈D w Si ) ∈ Lie S is obtained by taking w d , substituting the monomial w Si for the variable corresponding to i ∈ D, and finally replacing each letter c s with s ∈ S i by the letter c φ(s) .
If w is a multilinear monomial in letters c 1 , . . . c n , then sends w ⊗ v 1 ⊗ · · · ⊗ v n to w(v 1 , . . . , v n ). 3. 4. The Algebraic Branching Rule. We will now explain an algebraic branching rule for the Σ n -module Lie n , which will follow from our later topological Theorem 5.10 by applying homology.
Fix a decomposition n = n 1 + . . . + n k . Let S = n and g : n → k be order-preserving with is an n-labelled weak Lyndon word with period d. The different copies of u in w induce a partition n = Si∈dw S i via the labelling. Here d w denotes the set of classes of this partition. It has cardinality d, but no preferred identification with d = {1, . . . , d}. For each S i ∈ d w , we can restrict the labelling f on w to an (S i , g| Si )-labelling f i on u and form a wordũ fi ∈ Lie Si by replacing each c j with j ∈ k by the corresponding c s for s ∈ S i . Let θ w : Lie dw → Lie n be defined as follows: given a multilinear monomial w d ∈ Lie dw in letters {c Si } Si∈dw , replace each c Si in w d withũ fi . The group Σ dw acts naturally on Lie dw . It acts on n (and hence Lie n ) by letting h ∈ Σ dw send x ∈ S i to the unique h · x ∈ h(S i ) for which x and h · x label corresponding letters in the respective copies of u. The map θ w is Σ dw -equivariant. The homomorphism θ w : Lie 2 → Lie 8 is defined follows. Suppose w 2 (c S1 , c S2 ) is a monomial representing an element of Lie d2 . Then w 2 is sent to . In particular our Σ 2 -equivariant homomorphism acts as follows on words of length 2: Summing the homomorphisms θ w over all labelled weak Lyndon words, we obtain a homomorphism θ w : w∈B w n (n1,...,n k ) Lie dw −→ Lie n . The group Σ n1 × . . . × Σ n k acts on the left: given a permutation h and a word w d ∈ Lie dw , define h · w d ∈ Lie d h·w by replacing the letter c Si corresponding to S i ∈ d w by the letter c h(Si) corresponding to h(S i ) ∈ d h·w . As before, h · w is the labelled weak Lyndon word obtained from w by precomposing the labelling with h −1 (cf. Definition 3.6). The above morphism is equivariant.
Example 3.12. In the above Example 3.11, we take h = (124)(68). Then h · w is given by The following will be proven on p.42 by applying homology to Theorem 5.10:

The Algebraic Hilton-Milnor Theorem.
In topology, the Hilton-Milnor theorem is a homotopy decomposition of the space ΩΣ(X 1 ∨. . .∨X k ) as a weak product indexed by basic monomials in k variables X 1 , . . . , X k . Equivalently, one can read the theorem as a homotopy decomposition of the pointed free topological group generated by X 1 , . . . , X k , where X i are connected spaces.
The free Lie algebra over Z (in the sense of Definition 3.1) on a direct sum has an analogous decomposition, which we will call the algebraic Hilton-Milnor theorem. There are differences between the two versions: the algebraic version gives an isomorphism, rather than an equivalence, and it does not require a connectivity hypothesis. In our later Corollary 5.13, we will in fact generalise the algebraic Hilton-Milnor theorem to spectral Lie algebras.
We fix free abelian groups V 1 , . . . , V k and an order-preserving map g : Every Lyndon word u ∈ B(m 1 , . . . , m k ) has a unique (m, g)-labelling for which the labels of each letter c i increase from left to right in u. Writeũ for the corresponding resolution (cf. Definition 3.7).
We define a map of abelian groups β u : V ⊗m1 Using the algebraic branching rule in Lemma 3.13 , we can deduce the following result: obtained by taking the direct sum of all mapsβ u is an isomorphism.

Proof.
We construct a commutative square The vertical isomorphisms are given by our operadic description of free Lie algebras. The top map on the summand corresponding to m 1 , . . . , m k , u ∈ B(m 1 , . . . , m k ), d ≥ 1 is obtained as follows.
First, apply Σ dm1 × ... × Σ dm k -orbits to the following map induced by algebraic branching for Lie n : ⊗n . A routine diagram chase shows that the square in question commutes. The top horizontal map is an isomorphism by the algebraic branching rule in Theorem 3.13.
Remark 3. 16. There is a corresponding decomposition for isotropic Lie algebras, which is folklore and can, for example, be deduced without effort from Example 8.7.4 in Neisendorfer's book [Nei10]. 3.6. The Spectral Lie Operad. There is a close connection between the Lie representation and the cohomology of the partition complex: given a finite set S, there is an isomorphism of where sgn S denotes the sign representation. This was first proven by H. Barcelo [Bar90] and later generalised, and perhaps simplified, by Wachs [Wac98]. Salvatore [Sal98] and Ching [Chi05] have subsequently endowed the collection of spectra {Map Sp (S 1 , (S 1 ) ∧S ) ∧ D(Σ|Π S | )} S with the structure of an operad in a way that is compatible with the operad structure on {Lie S } S .
We will now explain these relations. Let P S be the poset of partitions of S, so that Π S = P S − {0,1}. The group Σ S acts by letting a permutation h ∈ Σ S send a partition i S i to the partition i h(S i ).
For |S| = 1, we model Σ|Π S | = S 0 by the simplicial 0-circle. The set of k-simplices of the simplicial model for Σ|Π S | is given by by the set of all chains starting in0 and ending in1 together with an additional basepoint * .

In order to prove Lie
and construct the operadic structure maps, we recall the language of weighted trees from [Chi05]: 17. A (rooted) tree consists of a finite poset T containing a unique minimal element r, the root, and another element b > r with b ≤ t for all t ∈ T − {r}, such that: (1) If u, v, and t are any elements of T with u ≤ t and v ≤ t, then u ≤ v or v ≤ u.
(2) If t, u are elements in T with t < u and r = t, then there exists a v with t < v and u v. Maximal elements are called leaves. Elements which are neither roots nor leaves are internal nodes. An edge is an inequality (u < v) such that there does not exist a t with u < t < v. Given t ∈ T , write i(t) for the set of incoming edges (t < v). A planar structure on T is given by a total order on each of the sets i(t) of incoming edges. The tree T is binary if every internal node has two incoming edges. Given a set S, an S-labelling on T is a bijection ι from S to the set of leaves of T .
A weighting on T consists of an assignment of nonnegative numbers to all edges of T such that the "distance" from the root to any leaf is 1. Write w(T ) ⊂ [0, 1] {edges of T } for the space of all weightings and ∂w(T ) for the subspace of weightings which have at least one vanishing weight.
Weighted S-labelled trees can be used to model the space Σ|Π S | . Starting with the simplicial model introduced above, we observe that points in Σ|Π S | are either equal to the basepoint or can be represented by pairs of the form • Given a chain of partitions σ ] of S, we let T σ be the set of equivalence classes of all partitions x i that are not classes in the preceding partition x i−1 (here x −1 has no classes by convention), partially ordered under refinement from coarsest to finest, together with an additional minimal element r, the root. This poset is a tree and it carries a canonical S-labelling. • Given (t 0 , . . . , t k ) ∈ [0, 1] k with t 0 + . . . + t k = 1, we give T σ the unique weighting for which the distance from the root r to each class in x i is precisely t i + . . . + t k .
Clearly, different weighted S-labelled trees can represent the same point in the space Σ|Π S | . In particular, any tree with weight 0 on a leaf edge or the root edge represents the basepoint.
Given a permutation h ∈ Σ S and an S-labelled weighted tree T , we define a new S-labelled weighted tree h · T by precomposing the labelling function from S to the set of leaves with h −1 ; in other words, we replace each label s ∈ S by h(s). This gives a concrete description of the Σ S -action on Σ|Π S | induced by the Σ S -action on the poset of partitions Π S .
In order to define spectral Lie algebras, we need one further ingredient from [Chi05]: Definition 3.18. Assume that we are given a bijection of finite sets φ : The ungrafting map u φ : Σ|Π S | −→ Σ|Π D | ∧ Si∈D Σ|Π Si | has the following two properties: (1) If we can find "partitioning edges" {(v − i < v i )} i∈D such that for each i ∈ D, the leaf labels above v i are given by φ(S i ), then we create a tree T by removing all vertices and edges from T which lie above the upper vertex v i of any partitioning edge. Weight T by giving "nonpartitioning" edges the same weights as in T and then scaling the weights of the partitioning Furthermore, for each i ∈ D, we create an S i -labelled weighted tree T i from T by only taking those vertices and edges in T which lie above the lower vertex v − i of the i th partitioning edge (v − i < v i ) and then simultaneously scaling the weights inherited from T appropriately. The ungrafting map u φ sends T to the point represented by (T , {T i } i∈D ).
(2) If, on the other hand, such partitioning edges cannot be found, then the ungrafting map u φ sends an S-labelled weighted tree T to the basepoint. Lie S1 We endow Lie with the structure of an operad as follows: given a bijection φ : The top map is simply given by reordering factors. The left component of the lower map is given in an evident manner by "plugging in". The right component of the lower map is obtained by applying D(−) to the "ungrafting map" u φ : Σ|Π S | −→ Σ|Π D | ∧ i∈D Σ|Π Si | defined above.
The spectral Lie operad Lie is a cofibrant replacement of Lie in the model category of constant-free operads in Sp ( [BB17], cf. also Proposition 3.6. (2) in [Cav]).
In particular, there is a map Lie → Lie of operads in Sp which is a trivial fibration in the functor category Fun(Fin ∼ = , Sp). The spectral Lie operad is Σ-cofibrant, which means that it is cofibrant in the projective model structure on Fun(Fin ∼ = , Sp). This implies that the associated monad X → n Lie n ∧ Σn X ∧n on Sp preserves weak equivalences.
We let Alg Lie (Sp) be the ∞-category of algebras for the induced monad Free Lie on the underlying ∞-category Sp of Sp. Its objects are called spectral Lie algebras. The induced monad on the homotopy category hSp is simply given by X → n Lie n ∧ hΣn X ∧n .

3.7.
Algebraic vs. Spectral Lie Operad. We will construct a Σ S -equivariant isomorphism show that it is compatible with the operadic structures. The existence of such an isomorphism was established abstractly via Koszul duality in Example 9.50 of [Chi05] -we shall give an explicit construction. For the rest of this section, we will work in the homotopy category hSp of spectra. Here Lie S and Lie S are canonically isomorphic.
Let T be a binary S-labelled tree whose partial order is denoted by for the open subspace of all points which can be represented (cf. p.26) by an S-labelled weighted tree that has nonzero weights on all edges and is isomorphic to T (as an S-labelled tree). Specifying a point in (Σ|Π n | ) T is equivalent to giving a weight in w(T ) all of whose components are positive.
There is a homeomorphism Fix a binary planar S-labelled tree T . We define the standard order v 1 < . . . < v 2|S|−1 on the non-root nodes of T by first listing the vertex above the root, then the non-root nodes of the left subtree T L (ordered by recursion), and finally the non-root nodes of the right subtree T R (ordered by recursion). Consider the subspace V ⊂ R 2|S|−1 cut out by the following equations The space V is an intersection of transversely intersecting hyperplanes (cf. Lemma 3.9 in [Chi05]) and hence has dimension dim(V ) = |S|−1. There is a natural identification w( In other words, we send a tuple (t 1 , . . . , t 2|S|−1 ) to the unique weighting on T with the property Passing to quotients by the respective boundaries gives rise to a homeomorphism S |S|−1 ∼ = w(T )/ ∂w(T ) .
Definition 3.20. Let T be a binary planar S-labelled tree. The map a T is defined as the composite The map b T : S |S| −→ (S 1 ) ∧S sends the k th coordinate in S |S| to the coordinate in (S 1 ) ∧S which corresponds to the label s k ∈ S of the k th leaf from the left in T .
Definition 3.21. The signature sgn(T ) of a binary planar tree T is sgn(T ) = 1 if T has no internal nodes. If T is obtained by first identifying the roots of a left subtree T L and a right subtree T R and then adding a new minimal element r, the root, then we set sgn(T ) = (−1) (# internal nodes of T L )·(# leaves of T R ) · sgn(T L ) · sgn(T R ). Alternatively, sgn(T ) is the sign of the shuffle permutation of the non-root nodes of T required to transform the standard ordering into the ordering which first lists all leaves and then lists all internal nodes, where these two sets are internally still ordered as in the standard order.
Every multilinear monomial w in letters {c s } s∈S canonically gives rise to a binary planar S-labelled tree T w : if w = c s is a letter, then T w has a root and a leaf labelled by s. If w = [u, v], then T w is obtained by first identifying the roots of T u and T v , then adding a new minimal element, and finally placing T u to the left of T v . We can now attach homology classes to words: Given a multilinear monomial w in letters {c s } s∈S , define a map γ w in hSp as We define x w ∈ H 0 (Lie S , Z) by pushing forward the fundamental class in H 0 (S 0 , Z) along γ w .
If h ∈ Σ S and w ∈ Lie S is represented by a multilinear monomial, we observe that γ h·w = Lie h •γ w .
The following diagram commutes in hSp: Proof. We will first explain the following two squares: We begin with the left square. Its top horizontal map is given by tree ungrafting (cf. Definition 3.19). The lower horizontal map is induced by the shuffle permutation of the internal nodes of T w(v1,...,v d ) which changes the standard order to the order that firsts lists the |d| − 1 internal nodes , and so on. Since ungrafting preserves the ratios appearing in the definition of the maps a T above, the left square commutes We move on to the right square above. Its lower map is canonical. The top horizontal map rearranges the various S |Si| 's so that the corresponding trees T vi appear from left to right in T w(v1,...,v d ) . Applying D(−) to the first square and S −1 ∧ (−) to the second, smashing both, and rearranging the terms gives a commutative square Assume now that the letters c 1 , . . . , c d appear in the order c σ(1) , . . . , c σ(d) in the word w . The top path S 0 → S 0 of this diagram is then homotopy equivalent to the map which shuffles negative spheres other than the leftmost S −1 to the right (while preserving their internal order). The sign of this map is equal to the sign of a corresponding permutation τ of the notes of T w(v1,. ..,v d ) : internal nodes corresponds to a copies of S −1 and leaves correspond to copies of S 1 .
We can now observe that sgn( ...,vn) ) since both are equal to the sign of the permutation of the non-root notes of T w(v1,. ..,vn) which moves all leaves to the left while preserving the internal standard order of leaves and internal nodes.
Since this cycle has even sign, the induced maps 1, σ, σ 2 : S 3 → S 3 are in fact all homotopic, and it therefore suffices to prove that D(Σ|Π 3 |) This fact was proven by the second author and Antolín-Camarena and is written in Proposition 5. 2. of [Cam16] Proposition 3.23 and Corollary 3.24 imply that the assignments w → γ w → x w from Definition 3.22 descend to Σ n -equivariant homomorphisms Lie n → π 0 (Lie) → H 0 (Lie, Z) -our later branching rule implies that they are in fact isomorphisms. Using Proposition 3.23, we deduce: Hence, the homology and homotopy groups of spectral Lie algebras form graded Lie algebras, i.e. Z-graded abelian groups g * with a bilinear product [−, −] : g i × g j → g i+j such that for all homogeneous u, v, w, the graded antisymmetry relation [u, v] Equivalently, graded Lie algebras are algebras over the algebraic Lie operad {Lie S } in the category of Z-graded abelian groups (endowed with the Koszul symmetric monoidal structure).
An easy variation of our arguments establishes the corresponding statements for the shifted spectral Lie operad (whose n th term is given by D(Σ|Π n | )) and shifted Lie algebras. 3.8. Collapse and Ungrafting. Let x be a proper nontrivial partition of the finite set S given by S = Si∈dx S i , where d x denotes the set of classes of x. We observe evident isomorphisms of posets (P S ) [0,x] ∼ = Si∈dx P Si and (P S ) [x,1] ∼ = P dx . We consider the following two maps: .
Proof. We begin by describing the map ν x explicitly. Let T be a weighted S-labelled tree in Σ|Π S | . Assume that cutting T along edges this is precisely what it means for T to lie in the interior of a simplex corresponding to a chain of partitions [. . .
For each S i ∈ d x , we form an S i -labelled weighted tree T i by restricting to the corresponding component above the "m-line", adding a root if necessary, and then rescaling the weights. Form a weighted d x -labelled tree T by first restricting to the component below the "M -line", then adding leaves where necessary, and finally rescaling the weights. The collapse map ν x from Section 2.10 then sends T to (T , The ungrafting map u x and the collapse map ν x naturally factor through maps We shall now produce an explicit homotopy from ν x to u x (and hence also from ν x to u x ).
Given an S-labelled weighted tree T ∈ (Σ|Π S | ) x , we again write e i for the length of the edge Given some t ∈ [0, 1], we define a new S-labelled weighted tree H x (t, T ) by taking the tree T and: • Scaling the weights of edges lying above the node v i by a factor of (1 − t) • Scaling all edges below some v − i by a factor of (1 − t) + tM 1 ≤ 1. An easy check shows that summing up the numbers in H x (t, T ) along an increasing path from the root to any leaf gives 1. To see that we have defined a valid weighting for all t, we need to show that all weights are nonnegative. This is clear for the first and last case in the definition of H x (t, T ). For the middle case, it suffices to prove the claim for t = 1. We observe that 1−( 1−m1 is readily checked to be a well-defined and continuous function The function K x can be continuously extended to a function Using notation from above, we let e n i , f n i , a n , b n , m n 1 , M n 1 , be the various numbers attached to T n . In H x (t n , T n ), the distance M n from the root r to the lowest vertex v i is given by After possibly decomposing the sequence into subsequences, we may assume without restriction that one of the following cases holds true: (1) m n ≥ M n for all n.
(2) m n < M n for all n and the sequence f n i tends to 1 for some i. (3) m n < M n for all n, there is a κ > 0 with f n j < 1 − κ for all n, j, and e n i −→ 0 for some i. (4) m n < M n for all n, there is a κ > 0 with f n j < 1 − κ and κ < e n j for all n, j, and the length r n of the root edge of T n tends to zero. (5) m n < M n for all n, there is a κ > 0 with f n j < 1 − κ and κ < e n j for all n, j, and the length of the leaf edge of T n labelled by s ∈ S i tends to zero.
. For case (1), we observe that the tree K x (t n , T n ) is constantly equal to the basepoint.
For case (2), we note that the i th leaf-edge of T n has length 1− For case (3), we check that the length of the root edge of the i th tree T n i is bounded above by For case (5), we use again that 1 − m n ≥ κ to see that the length of the leaf edge in T n i labelled by s tends to zero. This concludes the proof of the existence of the continuous extension K x .
, we note that for T a weighted S-labelled tree in (Σ|Π S | ) x , our above observations show that all nodes v i (in the above notation) lie above the "M 1 -line" in H x (1, T ) (with at least one on this line). Similarly, all nodes v − i lie below the "m 1 -line" (with at least one having distance m 1 from the root). We can then read off the identification K x (1, T ) = u x (T ) from our explicit description of ν x in the beginning of this proof. Finally, the equivariance claim is evident by the symmetry of the definition of H x . 3. 9. Binary Chains. Let S be a finite set. We will analyse collapse maps for the following chains: 27. An increasing chain of partitions σ = [x 0 ≤ · · · ≤ x r ] of S is said to be binary if for all i ≥ 0, each equivalence class in x i is the union of at most two equivalence classes in x i−1 . For i = 0, this means that each component of x 0 has at most two elements. However, note that we do not require x r to only have two equivalence classes.
Binary Chains of Length One. We begin our analysis with the following simple observation: We can describe the induced homeomorphism Σ|( y] whose coordinate at n ∈ N [x,y] is the distance from the root to the maximal node lying under all elements of n. ] be a binary chain of partitions starting in0 and ending in1. Moreover, assume that we are given a planar structure on the S-labelled rooted tree T σ associated with this chain (cf. Section 3.6). Consider the map x r+1 ] ∼ = S |S|−1 defined by sending the coordinate corresponding to the i th internal node in standard order (on the set ∪ r+1 i=0 N [xi−1,xi] of internal nodes of T , cf. Section 3.7) to the i th coordinate on the right. We can link α σ to the map a Tσ which we have associated with the binary planar tree T σ in in Definition 3.20: Here T h·σ inherits a planar structure from T σ .

Proof.
We proceed by induction on r, the statement being trivially true for r = −1.
For r ≥ 0, we assume that the tree T σ is constructed by first identifying the roots of a left subtree T σ1 and a right subtree T σ2 and then adding an additional minimal element r, the root. For i = 1, 2, let S i be the set of labels of T σi . Without restriction, we may assume that S 2 is a singleton. Write for the binary chain of partitions of S 1 obtained by restricting the first r + 2 partitions of σ to S 1 . We consider the following two composite maps: Here α σ1 is the collapse map associated with the chain σ 1 , the map a Tσ 1 is defined in terms of the S-labelled binary planar tree T σ1 (cf. Definition 3.20), the symbol ν xr denotes the collapse map for the partition x r , and the map u xr is the ungrafting map with respect to x r (cf. Section 3.8).
Using our explicit description of collapse maps in Section 2.10, we see that the top composition agrees with α σ . The bottom composition evidently gives the map a Tσ .
By Proposition 3.26, there is a homotopy K xr from ν xr to u xr . By induction hypothesis, there is a homotopy L σ1 from α σ1 to a Tσ 1 . Setting L σ := (id ∧L σ1 ∧id)•K xr gives the desired homotopy.
General Binary Chains. We can now finally analyse the collapse map associated with general binary chains. Assume that σ = [x 0 < . . . < x r ] is a binary chain of proper nontrivial partitions of S such that x r partitions S as S = Si∈dx r S i , where d xr is the set of classes of x r . Write x i j for the partition obtained by restricting x j to the set S i .
Suppose that for each i, we have chosen a planar structure on the S i -labelled binary tree T σi associated with the binary chain σ i obtained from [0 i ≤ x i 0 ≤ · · · ≤ x i r ≤1 i ] by deleting repeated partitions. Here0 i and1 i denote the minimal and maximal partition on the set S i . We write α σi : Σ|Π Si | → S |Si|−1 for the corresponding collapse map described on p. 31.
Combining the two homotopies from Proposition 3.26 and Proposition 3.29, we deduce: and have given the binary trees associated with h · σ the planar structures induced by the planar trees T σi .
In the unsuspended case, the collapse map takes the form Here Σ −1 Si∈dx r Σ|Π Si | denotes a desuspension of the sphere Si∈dx r S |Si|−1 . The unsuspended collapse map interacts with the group action in the expected manner.

Complementary Collapse
We will now present an algorithm which collapses large subcomplexes of order complexes attached to lattices and thereby produces equivariant simple homotopy equivalences to wedge sums of smaller spaces. Fix a finite group G throughout this section. 4.1. A Reminder on Discrete Morse Theory. We review the basics of discrete Morse theory: • The relation ∼ is G-invariant, i.e. σ ∼ τ implies gσ ∼ gτ for all g ∈ G.
• Every equivalence class is either equal to {x} or has the form {σ − , σ + }, where σ − is a face of codimension 1 of σ + .
Such a matching is acyclic if there does not exist a chain σ − with n > 1 and all σ i distinct. The following statement is due to Forman [For98] and Freij [Fre09]:

Complementary Collapse against Points.
A finite G-lattice P is a G-poset whose underlying poset is a finite lattice, which means that every two elements x, y have a meet x ∧ y and a join x ∨ y. Fix a finite G-lattice P and write P := P − {0,1}, where0 is the least and1 the largest element of P.
Before stating complementary collapse in full generality, we will present a special case: Here N (P) −x ⊥ is the complex containing all simplices in N (P) which do not contain a vertex in x ⊥ .
Using Corollary 2.27 and Proposition 2.22, we immediately obtain: Theorem 4.5. If x ∈ P is a G-stable element for which x ⊥ is discrete, then there are simple homotopy equivalences of G-spaces Remark 4.6. The contractibility of N (P) −x ⊥ and the resulting branching rule is originally due to Björner-Walker [BW83] in the nonequivariant and Welker [Wel90] in the equivariant case. The collapsibility in the nonequivariant case also follows from work by Kozlov [Koz98] on nonevasiveness. 4. 3. Orthogonality Fans. We will now prove a generalisation of Theorems 4.4 and 4.5 in which the "reference simplex" x is allowed to vary across the poset. We introduce some natural notation: Given any chain σ and any element z, we let σ <z denote the subchains of σ spanned by all elements x in σ which satisfy x < z. We write σ < z if all elements x in σ satisfy x < z and define [σ < z] to be the chain obtained by adding z to σ. We define the subchain σ >z , the condition z < σ, and the chain [z < σ] analogously.
If F : F P → P is any function and y ∈ P, we can define two new functions The variation of the reference simplex will be parametrised by the following structure: Definition 4.8. A list of functions (F 1 , . . . , F r ) from F P to P is an orthogonality fan if r = 0 or (1) F i is G-equivariant and increasing for all i (i.e. y ≤ F i (σ) for all σ ∈ F P and all y ∈ σ.)  9. A (possibly empty) chain σ = [y 0 < . . . < y k ] in P is said to be invisible for some orthogonality fan F = (F 1 , . . These two recursive definitions terminate. An F-invisible chain σ is said to be F-orthogonal if it is minimally invisible, i.e. if none of its proper subchains are F-invisible. Write σ ⊥ F and observe: Before stating the main theorem of this section, we unravel this definition in two cases of interest. (1) For t = 1, . . . , , we have The sequence [y i1 < . . . < y i ] is then orthogonal. We use the convention y i0 =0. For the empty sequence of indices ( = 0), the first two conditions are automatically satisfied whereas the third condition reads (3) For t = 1, . . . , + 1 and s = 1, . . . , m t , we have 4) For t = 1, . . . , + 1, we have F 2 ([0 < y i1 < . . . < y it−1 < y j t 1 < . . . < y j t m t ]) ∧ y it ∈ {y j t m t , y it }.
We use the conventions that y i0 =0 and y j t 0 = y it−1 for all t. Moreover, we set We can now state the two main theorems of this section:  Remark 4.15. One could use Theorem 4.5 and induction to prove the mere existence of the equivalence in Theorem 4.14. However, the equivalence produced in this way would not be easily accessible due to its inductive definition. On the other hand, the equivalence asserted in Theorem 4.14 is obtained by collapsing a large subcomplex all at once -the involved maps are therefore entirely transparent. Moreover, the collapsibility of this large subcomplex asserted in Theorem 4.13 does not follow from Theorem 4.14, but can be of independent interest.
Since the axiom (3) in Definition 4.8 is difficult to check, we introduce the following simpler notion: Definition 4. 16. Let G be a finite group and P a finite G-lattice with face set F P .
A function F : F P → P is called an orthogonality function if (1) F is G-equivariant and increasing (i.e. y ≤ F (σ) for all σ ∈ F P and all y ∈ σ.) (2) For all σ = [y 0 < · · · < y m ] ∈ F P and any z > y m , the following subposet is discrete: Orthogonality functions will provide us with many examples of orthogonality fans:  4. 4. Proof of Complementary Collapse against Fans. As before, let F = (F 1 , . . . , F r ) be an orthogonality fan on a finite G-lattice P. The following technical gadget will allow us to "scan" a chain σ in order to find out whether or not it is F-invisible: Definition 4.18. The orthogonality tree T F (σ) of a chain σ = [y 0 < · · · < y k ] in P is an empty or planar rooted tree whose nodes w are labelled by pairs (Z ∈ I w ) consisting of an interval I w = [y α , y ω ] in P and a Stab yα ∩ Stab yω -stable point Z ∈ I w .
The tree is defined by the following recursion: (1) If r = 0, then the tree is empty.
• We create the labelled rooted planar tree T F (σ) by declaring the root of L to be the left child and the root of R to be the right child of v. Note that L may be empty.
We call a vertex of the orthogonality tree a leaf if it has no children.
Example 4.19. Assume that F consists of two functions (F 1 , F 2 ). In the illustration in Example 4.12, we considered an invisible chain σ with y i1 < y i2 (indicated by long bars) and for which y j 1 1 < y j 1 2 , y j 2 1 < y j 2 2 < y j 2 3 , and y j 3 1 < y j 3 2 (indicated by short bars).
The orthogonality tree T F (σ) of σ is drawn on the right. The leftmost dotted node is The existence of the two other dotted nodes depends on similar conditions.  The next two statements follow by similarly straightforward inductions: Lemma 4.22. If σ ≤ τ is a subsimplex, then T F (σ) ≤ T F (τ ) is naturally a (labelled) subtree. Lemma 4.23. Fix a simplex σ with tree T F (σ) whose leaf w ∈ T F (σ) has label (Z ∈ [y α , y ω ]).
We can now prove complementary collapse for fans: Proof of Theorem 4.13. We define a G-equivariant perfect matching with fixed point Let w = w σ be the leftmost leaf of the orthogonality tree T F (σ) such that σ is not F-invisible at w (it exists by Lemma 4.21). Write (Z ∈ [y α , y ω ]) for the label of w. Since σ is not F-invisible at w, we have strict inequalities y α < Z < y ω . We use the convention that y −1 =0 and y k+1 =1.
Let i ≥ α be the largest index with y i ∧ Z = y α . Note that i < ω and observe that y i ∨ Z < y ω as otherwise w would not be a leaf. Let j ≥ i be maximal with y j ≤ y i ∨ Z. We have j < ω.
In the second case, we first observe that j > i as otherwise we would have y i = (y i ∨ Z) ∧ y i+1 which is absurd as y α = y i ∧ Z and y α < ((y i ∨ Z) ∧ y i+1 ) ∧ Z. We therefore remove a vertex y j in the open interval (y α , y ω ) and again conclude by Lemma 4.23 that the orthogonality trees of σ and σ − are equal . We therefore have i(σ − ) = i(σ) , j(σ − ) = j(σ) − 1, and T(σ) = T(σ − ).
We conclude that (σ − ) + = σ. We have thus defined a matching with fixed point F (0), and it is evidently G-equivariant.
To see that the matching is acyclic, assume for the sake of contradiction that we are given a cycle Here d t denotes the t th face map which forgets the t th element of a given chain.
Let (i s , j s , T s ) be the structure triple of σ s . We have observed above that the triple attached to σ + s is (i s , j s +1, T s ). By Lemma 4.22, we have T s+1 ≤ T s . Since the above is a cycle, the orthogonality tree T s must therefore be constantly equal to T, say. Let w be the leftmost non-invisible node of T with label (Z ∈ [y α , y ω ]).
We will now examine how i and j change as s increases. Fix s. By definition, the number i s+1 is the largest number with α < i s+1 (< ω) such that the i st s+1 vertex of σ s+1 = d ts (σ + s ) intersects Z in y α . Since σ s+1 = σ s , we know that t s = j s + 1. If t s ≤ i s then i s+1 = i s − 1. If t s > i s and t s = j s + 2, then σ s+1 is an upper simplex in the matching, a contradiction. If t s = j s + 2, then (i s+1 , j s+1 ) = (i s , j s + 1).
The function (i s , j s , T s ) hence cannot visit the same point twice -the above cycle cannot exist.

Restrictions
We will now use the full strength of complementary collapse against orthogonality fans to compute Young restrictions of the partition complex and parabolic restrictions of Bruhat-Tits buildings. 5. 1. Young Restrictions of the Partition Complex. Let P S be the lattice of partitions of the finite set S, ordered under refinement, so that Π S = P S = P S − {0,1}. Fix a map g : S → k and write C i = g −1 (i). Let Σ g = Σ C1 × · · · × Σ C k be the associated Young subgroup.
An orthogonality fan on the partition complex.
Given a chain of partitions σ = [y 0 < · · · < y m ] ∈ F P S and a class K of y m , we attach a word w K = w K (σ) ∈ F c 1 , . . . , c k in the free group on k generators to K: • If m > 0, we first use the chain [y 0 < · · · < y m−1 ] to attach a word to all classes in y m−1 .
We then let w K be the product of all words attached to y m−1 -classes which are contained in K, multiplied in ascending lexicographical order (where c 1 < · · · < c k ). For σ = [y 0 < · · · < y m ] a chain, we record the words attached to y m -classes in ascending lexicographic order as w a < w b1 < · · · < w bs , where possibly s = 0. Let A be the set of y m -classes whose associated word is the minimal w a . Define B 1 , . . . , B s in a similar manner and set B = ∪ i B i .
Define F 1 (σ) = F 1 (S, g)(σ) to be the partition obtained from y m by merging all classes in B. Define F 2 (σ) = F 2 (S, g)(σ) to be the partition obtained from y m by merging all classes in A.
Proof of Theorem 5. 2. By Lemma 4.17, it suffices to check that F 1 and F 2 are orthogonality functions in the sense of Definition 4.16. The functions F 1 , F 2 are clearly increasing and equivariant. To check axiom (3), let σ = [y 0 < · · · < y m ] be a chain of partitions in P S and let z > y m . Define the sets of y m -classes A and B 1 , . . . , B s as above. Let A ⊂ A be the collection of y m -classes which are merged with a class in B = ∪B i in z. Similarly, write B ⊂ ∪B i for the family of classes in B which are merged with a class in A in the partition z. For F 1 , we observe a natural injection obtained by sending a ∈ A to the unique f (a) ∈ B with a y f (a). We draw a suggestive example on the right. The subposet is therefore discrete and F 1 an orthogonality function. The statement for F 2 follows from a parallel argument.
Orthogonal Chains from Labelled Lyndon Words. We will now give a convenient description of the chains of partitions which are orthogonal to the fan F(S, g) from Theorem 5. 2. Let k * be the set of words of finite length in {1, . . . , k}, ordered lexicographically, and consider the free group F c α α∈k * . This free group inherits a lexicographic order based on the order on k * . A word w ∈ F c α α∈k * is called a weak Lyndon word if w ≤w for any cyclic rotationw of w.
Given w ∈ F c α , we define a sequence w 0 , w 1 , w 2 . . . by w 0 = w and w i = w i−1 for i > 0.
Example 5. 4 Proof. Given a weak Lyndon word w = c α1 . . . c αm , any rotation w of w lifts to a rotation w of w satisfying ( w) = w (since w is weak Lyndon). Observe that if w < w , then w < w.
Lemma 5. 6. If w = u d is a weak Lyndon word for which u is not a letter, then w = w.
Proof. If w = c α1 . . . c αm , then α 1 is minimal in {α 1 , . . . α k }. Hence either w = c d α1 or w = w. Corollary 5.7. If w is weak Lyndon, then w 0 , w 1 . . . stabilises in a word of the form c i α . We will use this construction to produce chains of partitions. First, we mildly extend terminology from Section 3.2. Fix a finite set S and a map g : S → k * . An (S, g)-labelling of a weak Lyndon word w = u d in letters {c α } α∈k * is a bijection between S and the letters of w such that each s ∈ S labels a symbol c g(s) . Two bijections give the same labelling if they agree after permuting the copies of u.
Let w be a weak Lyndon word in letters {c α } α∈k * together with an (S, g)-label f . For each i, we have a canonical function from S to the letters of w i . Let x i be the partition on S which identifies two points if they have the same image under this function. Let S xi = S/x i be the set of classes of x i and g xi : S xi → k * be the natural function recording the "type" α of the letter c α in w i corresponding to a given class in S xi . Then w i is naturally (S xi , g xi )-labeled, and x i , S xi , and g xi are well-defined for all i.
The illustration shows the chain of partitions attached to w = c 1 c 1 c 2 c 1 c 1 c 3 c 2 c 3 (cf. Example 5.4).
Definition 5. 8. The binary chain of partitions σ w attached to a labelled weak Lyndon word w is given by [x 0 < x 1 < . . . < x r ], where r is chosen maximal with x r =1.
Let now g : S → k be a map with fibres C i = g −1 (i) of size n i . We prove: Lemma 5. 9. The rule w → σ w gives a Σ C1 × · · · × Σ C k -bijection B w (S,g) (n 1 , . . . , n k ) Proof. We freely use notation from above. Equivariance is evident. For bijectivity, we induct on n = n 1 + . . . + n k . Assume without restriction that n 1 > 0. If n = n 1 , then the map identifies the unique weak Lyndon word c n 1 with the unique (empty) orthogonal chain. If n > n 1 , we can remove all occurrences of c 1 by iterated application of the reduction function (−) .
After having shown that the map B w (S,g) (n 1 , . . . , n k ) − → F ⊥ (S, g) is well-defined, we will now prove bijectivity. Sending w to (x m , [x 0 < · · · < x m−1 ], w m ) gives a Σ C1 × · · · × Σ C k -equivariant map (Sy,gy) (. . . , n 1 2 2 , . . . , n 1 2 k , n 12 , . . . , n 1k , n 2 , . . . , n k ) In order to prove that the map D is bijective, we produce an inverse E. Given y ⊥ F 1 ([0]), a chain [z 0 < · · · < z m−1 ] ⊥ F ≤y 2 , and a word w ∈ B w (Sy,gy) (. . . , n 1 2 2 , . . . , n 1 2 k , n 12 , . . . , n 1k , n 2 , . . . , n k ), we first produce a word w by replacing all letters of the form c 1 a i by the string c a 1 c i and observe that w is again a weak Lyndon word. We define an (S, g)-labelling of w: if s ∈ S maps to [s] ∈ S y with g y ([s]) = 1 i a, then we can pick a unique t ∈ [s] with g(t) = a. We then declare that s labels the (r + 2) nd letter from the right in the corresponding string c 1 c 1 . . . c 1 c a in w if r is minimal index with s ∼ t in the partition z r (using the convention z −1 =0). We have defined the inverse E.
A parallel construction can be carried out on chains of partitions: given any partition y ⊥ F 1 ([0]) of S, we write S y for the collection of equivalence classes of y and obtain a "type function" g y : S y → {. . . , 1 2 2, . . . , 1 2 k, 12, . . . , 1k, 2, . . . , k} from g. By Definition 4.9 of orthogonality, we obtain a Σ C1 × · · · × Σ C k -equivariant bijection F(S, Combining the two bijections for words and chains, we obtain the following commutative square whose lower horizontal map is bijective by induction: Topological Branching Rule. Fix a decomposition n = n 1 + . . . + n k . Let S = n and g : n → k be the unique order-preserving function with |g −1 (i)| = n i . Given any (n, g)-labelled weak Lyndon word w = u d ∈ B w n (n 1 , . . . , n k ), we have attached a binary chain of partitions σ w = [x 0 < . . . < x r ] ∈ F ⊥ (n, g) in Definition 5.8. The copies of u partition n into d sets via the labelling; we write d w for the set of classes of this partition. Observe that for each S i ∈ d w , the S i -labelled tree associated with the (S i , g| Si )-labelled Lyndon word u carries a planar structure. Using our analysis in Section 3.9, we can describe the collapse map associated with σ w as Combining this observation with Complementary Collapse in Theorem 4.14, our Lemma 5.9 concerning weak Lyndon words and orthogonal chains, and Remark 3.8, we obtain: Theorem 5. 10. For n = n 1 + · · · + n k , there are simple Σ n1 × · · · × Σ n k -equivariant equivalences Fix a decomposition n = n 1 +. . .+n k and let g : n → k be the increasing function with |g −1 (i)| = n i .
Let w = u d ∈ B w n (n 1 , . . . , n k ) be a weak Lyndon word with labelling represented by f . The various copies of u give rise to a partition x w , written as n = Si∈dw S i . For each S i ∈ d w , the Lyndon word u has an (S i , g| Si )-labelling represented by the restriction f i of f to S i . We write T i,w for the S i -labelled binary planar tree Tũf i corresponding to the Lie monomialũ fi obtained by resolving u.
An n-labelled weighted tree is said to be grafted from w if it has nonzero weights on all edges and we can find edges {(v − i < v i )} Si∈dw such that for each S i , the component of T above v − i is isomorphic to the S i -labelled tree T i,w . We write (Σ|Π n | ) (xw,Ti,w) for the open subspace of Σ|Π n | consisting of all points which can be represented by trees grafted from w.
Proof. Suppose a point in (Σ|Π n | ) (xw,Ti,w) is represented by the tree T having nonzero weights on all edges. We attach a weak Lyndon word in letters c 1 , . . . , c k to the nodes of T as follows. First, we assign the letter c g(s) to each leaf labelled by s ∈ n. Then, we label all non-leaves recursively: if the node t has incoming nodes t 1 , . . . , t k , we multiply the words attached to t 1 , . . . , t k in lexicographically increasing order and attach this product to t. The nodes v i are then characterised by the property that for all nodes above them, the attached words are strict Lyndon words, whereas for all nodes strictly below them, they are only weak Lyndon words. The word u is the label of any v i , and hence w = u d is determined.
We obtain a continuous Σ n1 × . .  We can now deduce the algebraic branching rule by applying homology: Proof of Lemma 3.13. Let w = u d ∈ B w n (n 1 , . . . , n k ). Write x w for the partition n = Si∈dw S i induced by the various copies of u in w via the labelling. For each S i ∈ d w , let f i be the (S i , g| Si )labelling on u obtained by restriction and write T i,w = Tũf i . By Corollary 3.25, we can obtain the map θ w appearing in the algebraic branching rule by applyingH 0 (−, Z) to the composite ϑ w given by Lie dw

Free Lie Algebras on Multiple Generators.
We will now describe free Lie algebras on direct sums of spectra X 1 , . . . , X k in terms of free Lie algebras on individual spectra.
Assume that we are given a strict Lyndon word w ∈ B(|w| 1 , . . . , |w| k ) of length |w| = i |w| i . Let S = {1, . . . , |w|} and write g : S → k for the unique order-preserving function with |g −1 (i)| = |w| i . We give w the (S, g)-label f which labels all copies of c i by elements in g −1 (i) in increasing order from left to right. Writew =w f ∈ Lie |w| for the Lie monomial obtained by resolving w.
In Section 3.7, we have constructed a map γw : S 0 → Lie |w| in the homotopy category of spectra.
Since Lie |w| is nonequivariantly equivalent to a wedge of 0-spheres, we can pick a morphism γw : S c → Lie |w| in Sp from a contractible space of representatives for γw. Here S c is a cofibrant replacement of the unit. Inducing up, we obtain a map Ind The final spectrum includes into n Lie n ∧ Σn (X 1 ∨. . .∨X k ) ∧n . Passing to underlying ∞-categories By Corollary 3.25 and a straightforward diagram chase, the effect of F w on homotopy is as explained on p.6.
Corollary 5. 13. Inducing up the maps F w and summing over all Lyndon words yields an equivalence of spectra Proof. It suffices to prove that for each (n 1 , . . . , n k ) with n = i n i , the component of w∈B k F w of multi-degree (n 1 , . . . , n d ) is an equivalence. In Sp, we can obtain this component by applying ..×Σn k to the following map of spectra with Σ n1 × . . . × Σ n k -action: ..×Σn k ) to the above map gives an equivalence. For this, it is in turn enough to check that the above equivariant map is a weak equivalence of underlying spectra, i.e. an isomorphism in the homotopy category. Here, this map can be obtained, after precomposing with an equivalence, by applying Map Sp (S 1 , (S 1 ) ∧n ) ∧ D(−) to the equivalence Remark 5.14. Exactly the same strategy can also be used to construct an equivalence of spectra for the shifted Lie operad whose n th term is given by D(Σ|Π n | ). Here, G w is the free map induced by the map of spectra G w , which is defined by replacing γw by D(a Tw ) in the definition of F w . The effect of G w on homotopy groups is given by the shifted variant of the rule described on 6.
An equivalence of spectra of the same form can also be deduced by applying Goodwillie calculus to the Hilton-Milnor theorem in topology as established by the first author and M. Kankaanrinta in [AK98]. However, in this approach, the arising maps are mysterious and it is not a priori clear how they interact with the Lie bracket. This obstruction is not present in our approach, where the interaction with the Lie operad is made entirely transparent by Proposition 3.23. 5. 4. Symmetry Breaking. Complementary collapse also gives an asymmetric version of Theorem 5. 10. Fix a partition S = A ∪ B 1 ∪ · · · ∪ B k corresponding to g : S → {1, . . . , k + 1}. Write x for the partition which identifies all points in B = ∪ i B i . Set F 1 ([0]) = x and F 1 (σ) =1 else. Let F 2 = F 2 (S, g) be as in the previous Section on p.39 for the orbits ordered as A < B 1 < · · · < B k .
Our argument in the last section shows that (F 1 , F 2 ) is an orthogonality fan. We deduce: Choosing a chain [z 1 < · · · < z < y] orthogonal to F amounts to the following data: • Choose a function f : A → B corresponding to a partition y = y f ⊥ x.
• Choose a set A 1 containing one f -preimage for each point in f (A) ⊂ B. We obtain a partition z 1 = z A1 by identifying each point in A 1 with its image under f . • Choose a set A 2 ⊂ A − A 1 containing one f -preimage for each point in f (A − A 1 ). We obtain a partition z 2 taking z 1 and merging all points in A 2 with their image under f .
Proceeding in this way gives all [z 1 < .. < z < y] ⊥ F -the claim follows by Theorem 4.14.
Remark 5. 16. Symmetry breaking can also be used to give an inductive proof of Theorem 5. 10. The disadvantage of this approach is that it is hard to describe the involved collapse maps.

Parabolic Restrictions of Bruhat-Tits Buildings.
Let V be a finite-dimensional vector space over a finite field k. Fix a flag A = [A 0 < · · · < A r ] with parabolic subgroup P A . We define an orthogonality fan (F ) of length 1 by F ( Choosing a flag B complementary to A with parabolic P B and intersecting Levi L AB = P A ∩ P B , we read off from Theorem 4.14: Lemma 5. 17. There is a simple P A -equivalence | BT(V )| ∼ = Ind P A L AB (Σ r r+1 i=0 | BT(gr i (B))| ). Here gr i (B) = B i /B i−1 for i = 1, . . . , r and we set gr 0 (B) = B 0 and gr r+1 (B) = V /B r .
Lemma 5.17 gives a new proof of a nontrivial result in modular representation theory. Write St n for the integral Steinberg module, i.e. the GL n (F p )-module given byH n−2 (BT(F n p ), Z). Corollary 5. 18. Let k = F q be a finite field and assume that R is any ring in which the number

Proof. Recall that whenever H ⊂ G are finite groups with |G| |H| invertible in R, then an R[G]-module M is projective if and only if its restriction to R[H] is a projective R[H]-module.
Let P A ⊂ GL n (F q ) be the subgroup of upper triangular invertible matrices stabilising the flag e 1 , e 1 , e 2 , . . . and write P ⊂ P A for the subgroup of matrices with all diagonal entries equal to 1. Since P intersects the Levi L AB corresponding to any complementary flag B trivially, Lemma 5.17 and the double coset formula imply that the restriction of St ⊗R to R[P ] is a free R[P ]-module. The claim follows since | GLn(Fq)| |P | = n k=1 (q k − 1) by a well-known counting argument.

Fixed Points
Given a subgroup G ⊂ Σ n , the subspace |Π n | G of G-fixed points carries a natural action of the Weyl group W Σn (G) = N Σn (G)/G. We will now provide an answer to the following question: Question. What is the W Σn (G)-equivariant simple homotopy type of |Π n | G ?
In order to present our analysis, we single out a special class of subgroups: Our result reduces the general case of the question raised above to the transitive situation: We again use the convention that if P is a poset with one element, then |P| ∧ X = X for any X.
Remark 6. 4. We could also view |Π n | G as a N Σn (G)-space or a C Σ d (G) n d -space. In this case, the above result implies equivalences Proof of Theorem 6.2 and Lemma 6. 3. Write x ∈ Π n for the partition of {1, . . . , n} into G-orbits.
We assume that G is nontrivial and that the action is not transitive. This implies that x =0,1.

Claim. The group G is isotypical if and only if
Proof of Claim. Let y ∈ x ⊥ be a partition with corresponding equivalence relation y on {1, . . . , n}. Given two elements a, b with a y b, we observe that for each c ∈ Orb G (a), there is a d ∈ Orb G (b) with c y d, and that this d must be unique as x ∧ y =0. We obtain an G-equivariant function Orb G (a) → Orb G (b) which by symmetry is bijective. Since y ∨ x =1, this implies that all orbits are isomorphic G-sets. Claim. The action of W Σn (G) on x ⊥ is transitive.
Proof of Claim. Given y, z ∈ x ⊥ , we define σ ∈ Σ n by setting σ(i) = j if and only if i, j lie in the same G-orbit and there is an g ∈ G with i y g(1) and j z g(1). A simple argument shows that this gives a permutation σ in if and only if i y j, which means that σ · z = y.
We can now deduce Theorem 6.2 and Lemma 6. 3. The Lemma follows immediately from Theorem 4.5 since x ⊥ = ∅. In order to prove Theorem 6.2, we take z ∈ x ⊥ to be the G-invariant partition with i z (i + kd) for all numbers i, k. The transitivity of the W Σn (G)-action on x ⊥ , Theorem 4.5, and Proposition 2.22 together imply the existence of a simple W Σn (G)-equivariant equivalence |Π n | G ∼ = Ind Proof of Lemma 6.5. We identify n with the G-set G/H. Let x be a G-invariant partition of G/H.
Then K is a subgroup of G. Indeed, suppose that we have eH x g 1 H x g 2 H. Multiplying the first equivalence by g 2 we obtain that g 2 H x g 2 g 1 H. Since g 2 H x eH, it follows that g 2 g 1 ∈ K. This shows that K is closed under multiplication. A similar argument shows that K is closed under taking inverses. Clearly, H ⊆ K. It is easy to see that a finer partition will lead to a smaller group K.
Conversely, let K be a group, H ⊆ K ⊆ G. Let x be the partition of G/H defined by the rule g 1 H x g 2 H if and only if g −1 2 g 1 ∈ K. It is easy to check that x is a well defined G-invariant partition, and that we have defined two maps between posets that are inverse to each other.
We can obtain specific contractibility results for certain subgroups: Let us suppose that d | n and d factors as a product d = d 1 · · · d l of integers d 1 , . . . , d l > 1 with l > 1. Consider the iterated wreath product Σ d1 · · · Σ d l as a subgroup of Σ n via the inclusions Proof. Without restriction, we may assume by Theorem 6.2 that d = n.
We can classify the isotropy groups of simplices in |Π n | which act isotypically: Lemma 6.7. Consider the action of Σ n1 × · · · × Σ n k on Π n . Let G ⊂ Σ n1 × · · · × Σ n k be an isotropy group of a simplex in Π n . Suppose that G acts isotypically on n. Then G is conjugate to a group of the form Σ d1 · · · Σ d l , where l ≥ 1, and d 1 · · · d l | gcd(n 1 , . . . , n k ). As usual, the group G acts diagonally on blocks of size d 1 · · · d l . The possibility that G is the trivial group is included.
Proof. Let σ = [x 0 < · · · < x i ] be a chain of partitions of n. Let G be the stabiliser of this chain in Σ n1 × · · · × Σ n k ⊂ Σ n . We will analyse the general form of G. The group G acts on the set of equivalence classes of each x j . Recall that x i is the coarsest partition of the chain. Let us say that two equivalence classes of x i are of the same type if they are in the same orbit of the action of G. This is an equivalence relation on the set of equivalence classes of x i . We say that there are s different types of equivalence classes of x i , and there are t 1 classes of type 1, t 2 classes of type 2 and so forth. It is easy to see that in this case G is isomorphic to a group of the form G ∼ = K 1 Σ t1 × · · · × K s Σ ts Here K j is the group of automorphisms (in Σ n1 × · · · × Σ n k ) of the restriction of the chain σ = [x 0 < · · · < x i ] to an equivalence class of x i of type j. We conclude that if G acts isotypically on n, then either G is trivial, or s = 1 and G ∼ = K 1 Σ t1 . Furthermore, if G acts isotypically on n, then K 1 has to act isotypically on a component of x i . The lemma follows by induction.
Homotopy theory often deals with computations "one prime at a time", and sometimes p-local questions about a given G-space X can be answered using only its fixed point spaces under pgroups. For example, Theorem 6.4 of [Dwy98] (cf. also Theorem A of [Web91] and Proposition 4.6.

of [ADL16]) specifies conditions under which the Bredon homology H
Br * (X, µ) of X with coefficients in a Mackey functor µ can be computed in terms of the fixed points of X under p-groups.
The fixed point spaces |Π n | P of partition complexes under p-groups are therefore of distinguished importance. The first author, William Dwyer, and Kathryn Lesh have shown in Proposition 6. 2. of [ADL16] that |Π n | P is contractible unless the p-group P is abelian and acts freely. Our Theorem 6.2 allows us to compute all non-contractible fixed point spaces of partition complexes under p-groups:

An EHP Sequence for Commutative Monoid Spaces
In this section, we will construct an EHP-like sequence for strictly commutative monoid spaces. This sequence will be a key tool in our subsequent study of strict Young quotients of the partition complex and our computation of the the algebraic André-Quillen homology of trivial square-zero extensions over F p for p odd. 7.1. Commutative Monoid Spaces and Simplicial Commutative Monoids. We begin by setting up the theory of commutative monoid spaces. Heuristically, it is often helpful to think of commutative monoid spaces as simplicial commutative rings over the field with one element.
We will use point-set models in order to facilitate our later definition of a new EHP-like sequence. Write Top * for the category of pointed (compactly generated weak Hausdorff) spaces, endowed with the usual Quillen model structure. Our definitions and results will be given for graded commutative monoid spaces -this will later allow us to decompose the EHP-like sequence into its graded components.
We fix an indexing monoid J with identity 0 ∈ J (in sets) once and for all. We write Top J * for the category of functors X (−) : J → Top * and give it the model structure whose fibrations, cofibrations, and weak equivalences are defined pointwise. Objects X ∈ Top J * will be called J-graded spaces, and we will often think of them as pointed spaces X together with a wedge decomposition X = j∈J X j .
The category Top J * is endowed with a symmetric monoidal structure given by Day convolution: The unit is given by S 0 concentrated in degree 0 ∈ J.
Definition 7.1. A J-graded commutative monoid space is a commutative monoid object in the symmetric monoidal category Top J * . We write CMon J for the category of J-graded commutative monoid spaces and let T be the monad on Top J * building the free such monoid. We will now unravel this definition. A J-graded monoid space R consists of a pointed space R together with a decomposition j∈J R j = R, a continuous multiplication map · : R × R → R, and a distinguished element 1 ∈ R 0 , such that the following three conditions hold true: (1) The multiplication is associative and commutative.
(2) The element 1 acts as a unit and the basepoint 0 satisfies 0 · x = 0 for all x ∈ R.
Here we glued together all basepoints 0 j of R j to a single point 0.
for f : X → Y pointwise a coproduct of generating cofibration of pointed spaces.
Strict pushouts of J-graded commutative monoid spaces are computed by relative smash products: Proposition 7. 3. If C ← A → B is a span of J-graded commutative monoid spaces, then We can place CMon in the framework of Batanin-Berger [BB17] and White [Whi17]. By (an evident graded generalisation of) Theorem 5. 4. in [Whi17] and Lemma 2. 3. in [Hov98], the category Top J * satisfies White's strong commutative monoid axiom and the monoid axiom, respectively. We say that a J-graded commutative monoid space is well-pointed if each of its components is a well-pointed space. Proposition 3.5 in [Whi17] implies that if M → N is a cofibration of J-graded commutative monoid spaces and M is well-pointed, then so is N . In particular, every cofibrant such monoid space is well-pointed. The following result appears as Theorem 4.18 in [Whi17] (generalising Theorem 3.1 in [BB17]):

This means that given a cofibration
gives rise to a weak equivalence h.
Relative left properness allows us to explicitly compute certain homotopy pushouts: Lemma 7.5. Assume we are given a pushout of J-graded commutative monoid spaces with j a cofibration and A, B, C well-pointed. Then the square is a homotopy pushout square.
Proof. We apply the usual argument for left proper model categories in the relative setting.
We introduce some variants. Let S 0 be the monoid with two elements in degree 0 (the unit of J).
Define the category CMon J,aug of augmented J-graded commutative monoid spaces as the overcategory CMon J /S 0 . It inherits a natural model category structure from CMon J such that the forgetful functor preserves fibrations, cofibrations, and weak equivalences.
We write CMon J,nu for the category of nonunital J-graded commutative monoid spaces, which are algebras over the monad T >0 (X) = n>0 X ∧n / Σn . Once more, this category is endowed with a model category structure whose fibrations and weak equivalences are defined on the level of spaces.
The augmentation ideal functor I (−) : CMon J,aug → CMon J,nu takes an augmented monoid space A → S 0 and assigns the preimage of the basepoint 0 ∈ S 0 . Every J-graded nonunital commutative monoid space X gives rise to a unital augmented monoid space S 0 ∨ X by adding a disjoint unit 1. The functors I and S 0 ∨ (−) do not assemble to an equivalence.
We can carry out a similar construction starting with the symmetric monoidal model category (sSet * , ∧, S 0 ) of pointed simplicial sets and follow Quillen to define a model structure on its category SCM J of J-graded simplicial commutative monoids. Weak equivalences and fibrations are again defined using the forgetful functor to sSet J * . Again, there is an augmented version SCM J,aug and a nonunital variant SCM J,nu . 7. 2. Extension of Scalars. Given a ring R and a commutative monoid space X, we will produce a simplicial commutative R-algebra R ⊗ X. Heuristically speaking, we extend scalars from F 1 to R. Consider the Quillen adjunctionF R : sSet * sMod R : U , where U is the forgetful functor andF R ( * → X) = coker(Free Mod R ( * ) → Free Mod R (X)) is the reduced levelwise free R-module construction. This above adjunction is monoidal. Passing to J-graded monoids, we obtain Quillen adjunctions SCM J SCR J R and SCM J,aug SCR J,aug R , whose constituent left and right components will also be denoted byF R and U , respectively. Here SCR J R denotes the model category of J-graded simplicial commutative R-algebras, i.e. commutative monoids in the category of functors J → sMod R (equipped with Day convolution) with fibrations and weak equivalences defined pointwise on underlying simplicial sets. The model category SCR J,aug R is a corresponding augmented variant. Note that no Koszul sign rule is imposed "in the J-direction". To see preservation of homotopy colimits, we observe that the functor Sing • : CMon J → SCM J is the right half of a Quillen equivalence, which implies that its right derived functor R Sing • preserves homotopy colimits. Since monoid spaces are fibrant, Sing • computes its own right derived functor and thus preserves homotopy colimits. The functorF R is left Quillen, and so its left-derived functor LF R preserves homotopy colimits. If Q → id is the cofibrant replacement functor on SCM J , we use thatF R preserves weak equivalences to see that LF R (X) ∼ =FR(QX) ∼ − →F R (X) is a weak equivalence. Hence, given D : In the last step, we used that the pointwise equivalence of diagramsF R • Q • D →F R • D induces an equivalence on homotopy colimits. 7. 3. Monoid Modules over Commutative Monoid Spaces. A monoid R ∈ CMon J is simply an algebra object in (Top J * , S 0 , ∧), and so there is a naturally defined theory of R-modules. We write MMod J R for the resulting category and call its members J-graded R-monoid modules.
We unpack the definition: a J-graded R-monoid module is a pointed space M together with a decomposition M = j∈J M j and continuous action maps · : R × M → M such that: (1) The action of R on M is associative.
(2) The element 1 ∈ R acts as the identity and 0 · m = 0 M for all m ∈ M .
We glued together all basepoints 0 M,j of M j to a single point 0 M .
We can also think of J-graded R-monoid modules as algebras over the monad X → R ∧ X.
Warning 7.8. If R is a topological ring, then the notion of an R-module is (evidently) not equivalent to the notion of a monoid module over the underlying commutative monoid space.
The category MMod J R carries a symmetric monoidal structure given by the relative smash product Again, cofibrations are retracts of transfinite compositions of relatively free maps A → B. In this context, these are maps which can be obtained as pushouts Given a J-graded R-monoid module M and a J-graded pointed space X, we endow the J-graded space M ∧ X with the structure of an R-monoid module by acting only on the M -factor.
We call a J-graded R-monoid module well-pointed if its underlying J-graded space has this property.
Lemma 7.11. Smashing with a cofibrant J-graded R-monoid module M preserves weak equivalences between well-pointed J-graded R-monoid modules.
Proof. Say that M has property ( * ) if it satisfies the conclusion of the theorem. Since retracts of modules with property ( * ) have property ( * ), we may assume that 0 → M is given by a transfinite Here the top map f i : A i → B i is a coproduct of generating cofibration of J-graded pointed spaces.
Since the filtered colimit of weak equivalences of J-graded pointed spaces is a weak equivalence, it suffices to check that if M i satisfies ( * ), then so does M i+1 .
We first observe that if X is a well-pointed J-graded space, then R ∧ X has property ( * ).
We will now prove that if M i satisfies ( * ), then so does M i+1 . Let S → T be a weak equivalence of well-pointed J-graded R-monoid modules. We consider the cube The three indicated arrows are weak equivalences by our previous argument and the induction hypothesis. The horizontal maps in the top square are given by smashing a cofibration of J-graded well-pointed spaces with the J-graded well-pointed spaces S and T respectively, hence these two maps are cofibrations of J-graded well-pointed spaces. Since pushouts of J-graded monoid modules are computed in J-graded pointed spaces, we conclude that the front and back square are homotopy pushouts of pointed spaces. This implies that the lower right map is a weak equivalence of J-graded spaces, hence of J-graded R-monoid modules.

Derived Smash Product of Monoid Modules.
We will now define an explicit model for the derived relative smash product of graded monoid modules. For this, it will be convenient to use ∆ k = {0 ≤ t 1 ≤ · · · ≤ t k ≤ 1 | t i ∈ R} as a model for the k-simplex. We think of ∆ k + as a J-graded pointed space concentrated in degree 0.
Fix a well-pointed R ∈ CMon J and two well-pointed monoid modules M, N ∈ MMod J R . For k ∈ N, we write points in the J-graded space ∆ k + ∧ M ∧ R ∧k ∧ N as For the sake of readability, we will often drop the second clause of this expression. We define: Definition 7.12. Given M, N ∈ MMod R , we define the space Here ∼ divides out by face and degeneracy maps. More explicitly, we quotient out by: If the unit S 0 → R is a cofibration of spaces and both M and N are well-pointed, then M ∧ h R N is well-pointed since the map S 0 → M ∧ h R N is the realisation of a levelwise cofibration between pointwise good J-graded simplicial pointed spaces. 7.5. Cofibrant Replacement of Monoid Modules. We can use the construction "M ∧ h R N " to define an explicit cofibrant replacement functor. For this, let R ∈ CMon J be a well-pointed graded commutative monoid space such that the unit S 0 → R is a cofibrant map of spaces.
Proposition 7. 13. Given a well-pointed J-graded R-monoid module M , the monoid module R ∧ h R M is cofibrant and the multiplication map R ∧ h R M → M is a weak equivalence.
An R-monoid module structure on D K can be defined using the leftmost copy of R. We obtain a transfinite composition of R-monoid modules R ∧ M = D 0 → D 1 → · · · → R ∧ h R M . Since transfinite compositions of cofibrations are cofibrations and the monoid module D 0 is cofibrant, it suffices to prove that all of the above maps are cofibrations in order to establish the first claim.
Let L K ⊂ R ∧K denote the subspace of all points {(r 1 , . . . , r k ) | r i ∈ R, ∃i s.t. r i = 1}. There is a natural pushout of J-graded R-monoid modules Since cofibrations are closed under cobase change, it is enough to prove that the top horizontal map is a cofibration of J-graded monoid modules. This map is obtained by starting with the cofibration of J-graded pointed spaces (∂∆ K ∧ R ∧K ∧ M ) ∂∆ K + ∧L K ∧M (∆ K + ∧ L K ∧ M ) → ∆ K + ∧ R ∧K ∧ M and applying the left Quillen functor R ∧ (−) -it is therefore a cofibration. The first claim follows.
To prove that the map R ∧ h R M → M is an equivalence, it suffices to show that this holds true for the underlying map of spaces. We note that this map is obtained from the augmented simplicial space X • : Bar • (R, R, M ) → M . The unit of R gives rise to a backwards contracting homotopy S : X n → X n+1 . This implies that we have a weak equivalence after applying hocolim ∆ op . Since S 0 → R is a cofibration, this homotopy colimit is computed by the geometric realisation.
, we see that the derived smash product deserves its name. 7.6. Derived Pushout of Commutative Monoid Spaces. We construct the derived pushout: Definition 7.14. If B ← A → C is a span of well-pointed J-graded commutative monoid spaces, we endow the space B ∧ h A C with a multiplication satisfying . . , a n+m , c 1 c 2 for any disjoint union {1 < · · · < m + n} = {i 1 < · · · < i n } {j 1 < · · · < j m }.
Observe that this multiplication is well-defined, commutative and associative.
Proposition 7. 15. Assume we are given a span of well-pointed J-graded commutative monoid spaces B ← A → C and that the unit of A is a cofibration. The square is a pushout, and in fact a homotopy pushout in CMon J .
Proof. It is evident hat the square is indeed an ordinary pushout square. In order to prove the second claim, we factor (A w − → B) as a cofibration followed by a trivial fibration in CMon J . The fact that A is well-pointed and A g − → B is a cofibration implies that B is well-pointed. The space A ∧ h A C also has this property. We obtain a diagram The upper square is a homotopy pushout by Lemma 7.5 since A → B is a cofibration.
The (underlying) map of A-monoid modules B ∧ h A C → B ∧ h A C can be obtained by smashing the weak equivalence B → B of well-pointed A-monoid modules with the cofibrant J-graded A-monoid module (A ∧ h A C). This implies by Lemma 7.11 that B ∧ h A C → B ∧ h A C is a weak equivalence of A-monoid modules and hence also of graded commutative monoid spaces.
We now turn to augmented commutative monoid spaces. As explained in Lemma 1. 3. of [Hir15], pushouts are computed in monoid spaces.
Definition 7. 16. The suspension Σ ⊗ A ∈ CMon J,aug of an augmented well-pointed J-graded Remark 7. 17. We obtain a model for the suspension in the pointed ∞-category of augmented strictly commutative monoid spaces.
We will now single out a class of monoid spaces which can be studied well via their homology: Definition 7. 19. An augmented monoid space R ∈ CMon J is said to be simply connected if it is well-pointed, its fibre over 0 is simply connected in each degree, and its fibre over 1 is a point concentrated in degree 0.
These monoids satisfy a desirable closure property: Lemma 7. 20. Let B f ← − A g − → C be a span of simply connected monoids in CMon J,aug and assume the unit of A is a cofibration. Then (our model for) B ∧ h A C is also simply connected.
We define a simplicial J-graded subspace X • ⊂ Bar • (A, B, C) = B ∧ A ∧• ∧ C as the fibre over 0. We also have the constant simplicial subspace S 0 The fibre over 1 is therefore clearly a point. Write A = A 0 ∨ S 0 , B = B 0 ∨ S 0 and C = C 0 ∨ S 0 for A 0 , B 0 , C 0 the fibres over 0. Then X n is a wedge sum of the pointwise simply connected J-graded spaces (B 0 ) ∧i ∧ (A 0 ) ∧j ∧ (C 0 ) ∧k with j ∈ {0, . . . , n}, i, k ∈ {0, 1}, and i + j + k > 0. Hence X n is itself pointwise simply connected. The J-graded space |X • | is therefore a homotopy colimit over the contractible category N (∆ op ) of pointwise simply connected J-graded spaces and therefore itself pointwise simply connected.
Definition 7.21. We call a sequence A → B → C of J-graded augmented commutative monoid spaces a cofibre sequence if the following square is a homotopy pushout: An analogous definition can be made for augmented J-graded simplicial commutative rings.
We can detect cofibre sequences on simply connected monoid spaces algebraically: Lemma 7. 22. Let A − → B → C be a sequence of simply connected monoid spaces in CMon J,aug for which Z ⊗ A → Z ⊗ B → Z ⊗ C is a cofibre sequence in SCR aug . Then A → B → C is a cofibre sequence of J-graded augmented commutative monoid spaces.

Proof.
We take the cofibre of the first map, i.e. the homotopy pushout of f against A → S 0 : The original sequence is a homotopy cofibre sequence if and only if k is a weak equivalence of J-graded spaces. We apply the reduced integral chains functor Z ⊗ (−) and obtain a diagram As Z⊗(−) preserves homotopy colimits (cf. Proposition 7.7), we see that Z ⊗ cofib(f ) cofib(Z ⊗ f ). Our assumption implies that Z ⊗ k is a weak equivalence. We deduce that k induces an isomorphism on integral homology. By Lemma 7.20, we can write cofib(f ) B ∧ h A S 0 = D 0 ∨ S 0 with D 0 the simply connected fibre over 0. Write C 0 for the fibre of C over 0. Since k is a map of augmented spaces, we have a map k 0 such that k = k 0 ∨ id : D 0 ∨ S 0 → C 0 ∨ S 0 . The map k 0 between simply connected spaces induces an isomorphism on integral homology and is therefore a weak equivalence. It follows that k is a weak equivalence. 7.7. Derived Pullbacks of Commutative Monoid Spaces. The theory of pullbacks is substantially more straightforward since fibrations are transported from J-graded pointed spaces.
Proposition 7. 24. Given a cospan B f − → A g ← − C of J-graded commutative monoid spaces, the following square is a homotopy pullback: for D the homotopy pullback in J-graded commutative monoid spaces. Since the forgetful functor U : CMon J → Top J * is right Quillen, its right derived functor preserves homotopy pullbacks. The functor U in fact computes its own right derived functor since all graded commutative monoid spaces are fibrant. Thus, U preserves homotopy pullbacks. After applying U , the dashed arrow is a weak equivalence of J-graded spaces since B × h A C computes the homotopy pullback in Top J * .
Definition 7. 25. Given a J-graded commutative monoid space A ∈ CMon J , the homotopy pullback S 0 × h A S 0 is denoted by Ω ⊗ A. Remark 7. 26. The monoid Ω ⊗ A models the loop object of A in the pointed ∞-category of augmented J-graded commutative monoid spaces.
Remark 7.27. If A → S 0 has fibres A 0 and A 1 over 0 and 1, then the underlying J-graded space of Ω ⊗ A is given by ΩA = ΩA 0 ΩA 1 . 7.8. The Suspension-Loops Adjunction. We will now set up a Quillen adjunction Σ ⊗ : CMon J,aug CMon J,aug : Ω ⊗ by defining the unit η A and the counit A by by It is readily verified that Σ ⊗ • Σ ⊗ η and Ω ⊗ • η Ω ⊗ are indeed given by the identity transformations and we have therefore defined an adjunction.
Proof. Let f : X → Y be a fibration or acyclic fibration in CMon J,aug . Then f is a fibration or acyclic fibration of underlying graded spaces. Write f 0 : X 0 → Y 0 and f 1 : X 1 → Y 1 for the maps induced on fibres over 0 and 1. The map Ω ⊗ (f ) is given on graded spaces by Ω(f 0 ) Ω(f 1 ) and hence a fibration or acyclic fibration of graded spaces. This implies that Ω ⊗ (f ) is a fibration or acyclic fibration of commutative monoid spaces. 7. 9. André-Quillen Homology for Commutative Monoid Spaces. We will follow Quillen's general approach in the context of J-graded commutative monoid spaces.
Definition 7. 29. The indecomposables functor Q : CMon J,nu → Top J * assigns to a J-graded nonunital commutative monoid space A the quotient space Q(A) = A/A · A Here A · A ⊂ A is the J-graded space consisting of all elements which can be decomposed into a product of two elements in A. We can also form square-zero extensions: Definition 7. 30. Given a J-graded space X, write X for the nonunital J-graded commutative monoid space obtained by declaring all products x · y of points in X to be equal to 0.
As expected, these functors form a Quillen adjunction Q : CMon J,nu > < Top J * : (−). The André-Quillen homology of an augmented monoid space is given by H Q * (A) := π * (AQ(A)). We give a formula for the André-Quillen chains of a graded commutative monoid space. Write T >0 for the monad building the free J-graded nonunital commutative monoid space. Then: Proposition 7.32. If A ∈ CMon J,nu is a nonunital commutative monoid space, then the André-Quillen chains of A are given by AQ(A) hocolim ∆ op (Bar • (1, T >0 , A)).
Proof. The J-graded nonunital simplicial commutative monoid space Bar • (T >0 , T >0 , A) → A has a contracting homotopy. Hence hocolim ∆ op (Bar • (T >0 , T >0 , A)) ∼ = A. Since the left derived functor LQ preserves homotopy colimits, we compute We will now see that André-Quillen chains for J-graded commutative monoid spaces behave well under "base-change" to ordinary rings. We fix an ordinary ring R and write Q : SCR J,nu Let Sym >0 R (X) = n>0 X ⊗ R n Σn be the free nonunital J-graded simplicial commutative R-algebra monad on J-graded simplicial R-modules. We can again give an explicit formula for the André-Quillen chains as AQ R (A) = hocolim ∆ op Bar • (1, Sym >0 R , A). We recall that reduced R-valued chainsC • (−, R) interact well with symmetric powers, i.e.
for all J-graded spaces X. André-Quillen chains therefore intertwine with extension of scalars: Lemma 7.34. There is a commutative square Given a nonunital J-graded commutative monoid space A, we computẽ Here we used thatC • (−, R) preserves homotopy colimits. 7. 10. An EHP Sequence in CMon. In this section, we will construct a new cofibre sequence for J-graded commutative monoid spaces. The "Hopf map" is difficult to define, and this is where our point-set approach becomes helpful. We recall the following notation: Definition 7. 35. Given a J-graded space X, the trivial square-zero extension of S 0 on X, written S 0 ∨ X ∈ CMon aug , is given by endowing the space S 0 ∨ X with the multiplication Algebraic and topological square-zero extensions interact well, i.e. R ⊗ (S 0 ∨ X) = R ⊕C • (X, R).
The monoid suspension Σ ⊗ of a trivial square zero extension has an explicit description: Proposition 7. 36. Given a J-graded pointed space X, there is a splitting of J-graded spaces Proof. This is evident from Definition 7.12 since points in X multiply to zero in S 0 ∨ X.
We will now proceed and define the Hopf map. Fix a J-graded pointed space X. We define a map of J-graded pointed spaces φ by x y Proposition 7.37. Given two points a 1 , a 2 ∈ S 1 ∧ X ∧2 , we have φ(a 1 ) · φ(a 2 ) = 0.
Proof. Write a i = (0 ≤ t i ≤ 1, x i , y i ) for i = 1, 2. Then: The last step follows as a coordinate is repeated and products of elements in X vanish in S 0 ∨ X.
We obtain a map φ : (S 0 ∨ ΣX ∧2 ) → Ω ⊗ Σ ⊗ (S 0 ∨ X) of J-graded augmented commutative monoid spaces and can define: Definition 7. 38. The Hopf map for J-graded augmented commutative monoid spaces is given by the adjoint H : More explicitly, H is given by the following map: x n , y n Proposition 7.39. There is a commutative diagram The top map is the weak equivalence We define the second map E in our EHP-sequence: Definition 7. 40. The map E : Σ ⊗ (S 0 ∨ X) → (S 0 ∨ ΣX) of J-graded augmented commutative monoid spaces given by collapsing all higher summands to the basepoint is called the Einhängung.
We record the following easy observation: Proposition 7.41. The following diagram commutes: Definition 7. 42. The EHP-sequence based at a J-graded space X is given by the following sequence of J-graded augmented commutative monoid spaces: Proposition 7. 43. The composite E • H factors through the zero object S 0 .
Proof. This follows immediately from our explicit description of the map H.
We can now prove the main theorem of this section. Let J = N be the monoid of nonnegative integers with addition.
Theorem 7. 44. For X = S n a sphere of even dimension n ≥ 2, thought of as an N-graded space placed in degree 1, the EHP-sequence is a homotopy cofibre sequence of augmented N-graded strictly commutative monoid spaces.
Proof. All three monoid spaces are simply connected. By Lemma 7.22, it therefore suffices to prove that this sequence is a cofibre sequence after extension of scalars to Z. We therefore need to check that the following sequence of simplicial commutative rings is a cofibre sequence: Here we made use of the weak equivalenceC Let D be the cofibre of the first map. By a computation originally due to Milgram, we have: Using Propositions 7.39 and 7.41, we see that Z ⊗ H and Z ⊗ E map elements to elements with "same name". There is a spectral sequence E 2 ] is evidently an isomorphism as well. This implies that D → Z ⊕ Z[n + 1] is a weak equivalence of simplicial commutative rings.
In this section, we will relate these quotients to the André-Quillen homology of commutative monoid spaces and simplicial commutative rings, describe the conditions under which they are wedges of spheres, use our EHP-like sequence and the branching rule to decompose them into simpler "atoms", and finally compute their homology with coefficients in Q and F p for any prime p.
We will now decompose the spaces C Lie (S a1 ∨ . . . ∨ S a k ) into atomic building blocks of the form C Lie (S ) = n≥1 Σ|Π n | ∧ Σn (S ) ∧n for an odd natural number.
From wedges of spheres to spheres: Given pointed spaces X 1 ∨ . . . ∨ X k , graded over Z k by placing X i in multi-degree e i , our Theorem 5.10 on Young restrictions gives an equivalence The d th piece in the summand C Lie (S 1+···+ k −1 ∧X ∧ 1 1 ∧· · ·∧X ∧ k k ) has multi-degree (d 1 , . . . , d k ). Corollary 8.1. If X i = S ai , then we obtain an equivalence of N k -graded spaces Hence, it suffices to study the value of C Lie on spheres to describe its behaviour on wedges of spheres.
From spheres to odd spheres: In order to reduce to the case of odd spheres (which is a key simplification for our subsequent cohomological computations), we need a conceptual understanding of the functor C Lie . The functor C Lie is closely related to square zero extensions in commutative monoids by the following observation: Lemma 8. 4. If X is a well-pointed J-graded space, then AQ(S 0 ∨ X) C Lie (X).

Proof. Write O nu
Comm for the reduced operad in Top * whose component at a finite set S is given by S 0 if |S| = 0 and by * if S = ∅. All structure maps are the identity. We can think of O nu Comm as an algebra object in the category SSeq(Top * ) of symmetric sequences in pointed spaces. Here the composition product serves as the monoidal structure.
There is a natural monoidal functor F : In the second equivalence, we have used the well-known identification between the simplicial set (Bar • (O nu Comm )) n and the simplicial model for Σ|Π n | explained in Section 2. 9.
Combining Lemma 8.4 with our EHP sequence for commutative monoid spaces in Theorem 7.44, we can reduce the study of C Lie evaluated on spheres to the study of C Lie evaluated on odd spheres: For X = S n an even sphere placed in degree 1 of J = N, there is a cofibre sequence of pointed More explicitly, for each d ∈ N, there is a cofibre sequence Here we use the convention that the space on the left is equal to the point if d is odd.
Proof. The first claim follows by applying AQ to the EHP-sequence in Definition 7.42 and using that AQ preserves homotopy colimits to commute suspension past the AQ functor.
The second claim for n = 0 follows directly by combining Theorem 4.3 in [Koz] with our Lemma 8.6 below. For n ≥ 2 even, we can again use that AQ preserves homotopy colimits to deduce that the EHP-cofibre sequence The result follows by applying Lemma 8. 4. We deduce the final concrete expression by passing to individual degrees.
We have therefore assembled the space C Lie (S 1 ∨ . . . ∨ S k ) from the simpler spaces C Lie (S ) for odd. This reduces the problem of analysing the space Σ|Π n | / Σn 1 ×···×Σn k to spaces of the form Σ|Π d | ∧ Σ d (S ) ∧d for ≥ 1 odd.
We will now provide the description of the space C Lie (S 1 ) which we had omitted in the proof above: 8. 2. The space C Lie (S 1 ). We compute the André-Quillen homology of the monoid S 0 ∨ S 1 : Lemma 8. 6. There is an equivalence of spaces This entails that in our computations, we may often assume that = 2 (which in particular implies that Σ −1 |Π d | ∧ Σ d (S ) ∧d is simply connected). First we need an auxiliary result: Lemma 8.7. Let Y := Σ d1 ×· · ·×Σ di be a non-trivial Young subgroup of Σ d , where d = d 1 +· · ·+d i . Let N be the normaliser of Y and let W be any group satisfying Y ⊆ W ⊆ N . Equip S d−1 with the standard action of Σ d . Then the orbit space S d−1 / W is contractible.

Proof.
We start by observing the well-known fact that the orbits space S d−1 / Σ d is contractible for d > 1. One way to see this is to identify S d−1 with |B d | , where B d again denotes the poset of proper, non-trivial subsets of the set d (Example 2.5).
of the simplicial nerve of B d by the action of Σ d is isomorphic to the nerve of the linear poset 1 < 2 < · · · < d − 1. In particular, its geometric realisation is contractible.
For the general case, we first observe the Y -equivariant homeomorphism By our assumption, at least one of the d j s is greater than one, and so the right hand side is a contractible space. Moreover, we claim that it is contractible as a N/Y -equivariant space. In other words, we claim that for every subgroup Again, at least one of the d j is greater than 1, so at least one of these factors is contractible, and therefore the whole space is contractible.
It follows that the orbit space S d−1 / Y / H is contractible for every H ⊂ N/Y . Finally, it follows that for every Y ⊂ W ⊂ N , the orbit space S d−1 / W is contractible.
Proof of Lemma 8. 6. The space S d−1 ∧ Σ d |Π d | is a pointed homotopy colimit of spaces of the form We claim that the space S d−1 / G is contractible for every G that occurs. From the claim it follows that the pointed homotopy colimit is contractible.
It remains to prove the claim. Let G be an isotropy group of |Π d | . Then either G = Σ d or G is the stabiliser group of a chain of proper non-trivial partitions of d.
In the first case, we have In the second case, suppose that G is the stabiliser of the chain of partitions [x 0 < . . . < x r ]. Here x 0 is the finest partition in the chains. Suppose x 0 has i equivalence classes of sizes d 1 , . . . , d i . Write Y ∼ = Σ d1 × · · · × Σ di for the group of permutations that leave the equivalence classes of x 0 invariant. Then G contains Y and is contained in the normaliser of Y , and S d−1 / G is contractible by Lemma 8.7. 8. 3. Homology of Strict Orbits. We proceed to examine the homology of strict Young quotients of the partition complex, or more generally of spaces of the form C Lie (X) for X a wedge of spheres.
We begin with a conceptual interpretation of the homology of the spaces C Lie (X). Given a ring R and a J-graded simplicial R-module M , we write R ⊕ M for the trivial square zero extension of R by M . Lemma 8.4 and Lemma 7.34 together imply: Theorem 8. 8. If X is a well-pointed J-graded space and R is a ring, then An aside on Operations. We shall briefly digress and explain why the R-valued singular cohomology classes of the spaces C Lie (S 1 ∨ . . . ∨ S k ) give rise to natural operations on the algebraic André-Quillen homology groups of simplicial commutative R-algebras.
Write Σ for the endofunctor on simplicial R-modules corresponding to the left shift on chain complexes. We observe that Lemma 7.34 and the standard adjunction between algebraic André-Quillen chains AQ R and trivial square-zero extensions together imply an identification Moreover, for any simplicial R-algebra A, we have an identification We therefore obtain maps We return to our main task of computing the homology of spaces C Lie (X) for X a wedge of spheres. Our combinatorial work in Section 4 and 5 has algebraic consequences. Indeed, we can apply H * (−, R) to Corollary 8.1 and deduce: Corollary 8. 9. There is a splitting ...,n k w∈B(n1,...,n k ) We find it noteworthy that this nontrivial theorem in algebra follows formally from our discrete Morse theoretic computations. For R = F 2 , this result has been established by Goerss [Goe90] by entirely different means.
We start by describing the homology of C Lie (S ) for S an odd sphere, and begin with the simplest case of rational coefficients: Lemma 8. 10. Let ≥ 1 be an odd integer. Then Proof. We give one of several possible proofs. It suffices to show that H to induce an isomorphism on rational homology. Therefore it is enough to prove the lemma with the homotopy orbit space replacing the strict orbit space. This was done in [AM99,Aro06]. For the sake of completeness we will sketch a proof. The space Σ −1 |Π d | ∧ hΣ d (S ) ∧d is a pointed homotopy colimit of spaces S d−1 hG , where G is an isotropy group of |Π d | . By an easy Serre spectral sequence argument, such a space is rationally trivial if is odd and G contains a transposition. It is clear that every isotropy group of |Π d | contains a transposition.
For R = F p with p a prime, the computation is significantly more difficult. The first author and Mahowald computed the homology of the homotopy orbits D(Σ|Π d | ) ∧ hΣ d (S ) ∧d for odd by using a bar spectral sequence whose input is the homology of extended powers as computed by Cohen-Lada-May [CLM76], Araki-Kudo [KA56], and Dyer-Lashof [DL62].
We can in fact run a related computation, the input of which is the homology of symmetric powers as computed by Dold [Dol58], Nakaoka [Nak57] [Nak59], and Milgram [Mil69]. We obtain: Theorem 9.1. Let ≥ 1 be a positive integer, odd if the prime p is odd. If n is not a power of p, then H * Σ|Π n | ∧ Σn S n , F p is trivial. If n = p a , then H * Σ|Π p a | ∧ Σ p a S p a , F p has a basis consisting of sequences (i 1 , . . . , i a ), where i 1 , . . . , i a are positive integers satisfying: (1) Each i j is congruent to 0 or 1 modulo 2(p − 1).
This theorem is proven in our final Section 9.
We will now use the EHP-sequence from Theorem 8.5 to express the homology of spaces C Lie (S ) with even in terms of the homology groups of C Lie (S j ) with j odd as computed in Theorem 9.1. We begin on the level of chains, where we can immediately deduce: Corollary 8.11. For j ≥ 0 even, there is a cofibre sequence of N-graded simplicial R-modules We analyse the effect of the EHP-sequence on (co)homology with coefficients in R = Q and R = F p : Theorem 8.12. Let n be an even natural number.
induces a short exact sequence of N-graded vector spaces on (reduced) homology and cohomology.
induces a short exact sequence of N-graded vector spaces on (reduced) homology and cohomology.
Proof. The rational statement is evident from Lemma 8. 10.
We read off the cohomological case for R = F 2 from Theorem 8.8 and Corollary B.2. in [Goe90]. The homological case follows by applying the universal coefficient theorem to all graded pieces. Now assume that R = F p with p an odd prime. We recall that the EHP-sequences decomposes into a wedge of cofibre sequences It will be enough to check the claim for each of them individually.
If d = p k , then the left hand side is contractible and the claim follows. If d = 2p k , Theorem 9.1 shows that the right hand side has vanishing F p homology and cohomology; the claim follows. If d is not of this form, both sides have vanishing F p -homology by Theorem 9.1.
Example 8. 13. We unpack the EHP-sequence at weight d = 2. Here, symmetric squares can be expressed in terms of real projective spaces (cf. [JTTW63]) and we in fact recover the cofibre We can combine our results to prove: Theorem 8.14. Fix integers 1 , . . . , k ≥ 0 and consider the homology group This group is given by the algebraic André-Quillen homology AQ R * (R ⊕ (Σ 1 R ⊕ . . . ⊕ Σ k R)) of the trivial square zero extension of R by generators x 1 , . . . , x k in simplicial degrees 1 , . . . , k .
For R = F p , the above homology group has a basis indexed by sequences (i 1 , . . . , i a , e, w), where w ∈ B k (n 1 , . . . , n k ) is a Lyndon word and e lies in {0, }. Here = 1 if p is odd and |w| is even or if |w| = 0. Otherwise, we have = 0. The sequence i 1 , . . . , i a consists of positive integers satisfying: (1) Each i j is congruent to 0 or 1 modulo 2(p − 1).
Proof. The identification with André-Quillen homology is a special case of Theorem 8.8.
We first examine the case R = Q. For k = 1, and 1 odd, this follows from Lemma 8. 10. For 1 even, it is implied by Theorem 8.12. For k > 1, the rational statement follows from Corollary 8. 9. Now let R = F p with p odd. If k = 1 and 1 ≥ 1 is odd, the statement follows from Theorem 9.1. If 1 is even (and thus = 1), the split short exact sequence asserted for p odd in Theorem 8.12 gives a direct sum decomposition By the "odd case", the first summand has a basis consisting of all sequences (i 1 , . . . , i a ) satisfying (1), (2), and the condition i a ≤ (p − 1)( 1 + 1). In light of (1), the condition i k ≤ (p − 1)( 1 + 1)" is equivalent to the condition i k ≤ (p − 1) 1 + 1. We therefore obtain all sequences with e = 0 appearing in the theorem. The homological degree of this sequence in (i 1 + . . . + i a + 1 + 1 + a) in H * (C Lie (S +1 ), F p ) and hence goes to an element of dimension (i 1 + . . . + i a + 1 + a) in H * (C Lie (S +1 ), F p ).
For R = F 2 , the second statement for k = 1 and 1 ≥ 1 follows directly from Theorem 9.1. For 1 = 0, we simply use Kozlov's theorem [Koz]. We deduce the case k > 1 from Corollary 8. 9. For p = 2, we obtain an independent proof of Goerss' computation (cf. [Goe90]) of the algebraic Andé-Quillen homology of trivial square-zero extensions over F 2 . Note that in this case, we do not make use of the EHP sequence.
We call a sequence (i 1 , . . . , i a , e, w) allowable with respect to Q or F p if it satisfies the conditions in the theorem above (here we use the convention that a = 0 in the rational case).
This concludes the computation of the homology of strict Young orbits of the partition complex, and we can finally answer the question we started with by setting 1 = . . . = k = 0: Corollary 8. 15. Let n = n 1 + . . . + n k .
The vector space H * (|Π n |/ Σn 1 ×...×Σn k , Q) has a basis consisting of all sequences (e, w ∈ B(m 1 , . . . , m k )) which are allowable with respect to Q and satisfy m i (1 + e) = n i for all i.
Proof. The space |Π n |/ Σn 1 ×···×Σn k is a wedge sum of spaces of the form Σ −1 |Π d | ∧ Σ d (S ) ∧d , where d divides gcd(n 1 , . . . , n k ). So it is enough to show that this space has no p-primary torsion for p > d. The cases d = 1, 2 can easily be checked "by hand". Suppose that d > 2 and p > d. To prove that the homology of Σ −1 |Π d | ∧ Σ d (S ) ∧d has no p-primary torsion it is enough to prove that its reduced homology groups with F p -coefficients are zero, and this follows from Theorem 9.1. 8. 4. Spherical Quotients. We will now provide an answer to the following elementary question: For which values of n = n 1 + . . . + n k is the quotient space |Π n |/ Σn 1 ×···×Σn k equivalent to a wedge sum of spheres?
For this, we will first carry out a detailed analysis of the spaces Σ −1 |Π p | ∧ Σp (S ) ∧p ∼ = |Π p | ∧ Σp S p−1 for p a prime. We will show that for many values of and p, these do not form wedges of spheres. This will allow us to conclude that the above quotients do not form wedges of spheres for many values of n i by our wedge decomposition in Proposition 8. 2. For the remaining values of n i , we do indeed find a wedge of spheres.
We will begin by examining the weight p component |Π p | ∧ Σp S p−1 and prove: Proposition 8. 17. Let p be a prime and a positive integer.
For p odd and odd, the quotient |Π p | ∧ Σp S p−1 is equivalent to the p-localisation of S p−1 / Σp .
For p odd and even, the space |Π p | ∧ Σp S p−1 is equivalent to the p-localisation of the homotopy cofibre of the quotient map S p−1 → S p−1 / Σp . For p an odd prime, the statement of this Lemma is subtle. In order to prove it, we will first analyse the space |Π p | ∧ Σp S p−1 in terms of homotopy coinvariants.
For this, recall that there is a Σ p -equivariant homeomorphism S p−1 ∼ = S −1 ∧ (S p−1 ) ∧ , where Σ p acts on S p−1 via the reduced standard representation, and acts trivially on S −1 . The fixed-points space (S p−1 ) Σp is homeomorphic to S −1 . Let S −1 → S p−1 be the inclusion of fixed points. The key to analysing |Π p | ∧ Σp S p−1 for a general prime p is the following proposition: Proposition 8. 18. The following is a homotopy pushout square Proof. The homotopy cofibre of the map S −1 → S p−1 is S ∧ (S (p−1)−1 ) + . Therefore, taking homotopy cofibres of the vertical maps we obtain the map We want to prove that this map is a homotopy equivalence. For this, it is enough to prove that the following map is an equivalence The Σ p -space S (p−1)−1 can be written as a homotopy colimit of sets of the form Σ p /G, where G is an isotropy group of S (p−1)−1 . Therefore, it is enough to prove that for every isotropy group G of S (p−1)−1 , the map |Π p | ∧ hΣp (Σ p /G) + → |Π p | ∧ Σp (Σ p /G) + is an equivalence.
Every isotropy group of S (p−1)−1 is contained in a group of the form Σ p1 × · · · × Σ p k , where p 1 + · · · + p k = p, k > 1, and p i > 0. For = 1, S p−2 is the boundary of the (p − 1)-simplex, and it is easy to see that the isotropy groups have this form. For > 1, we can argue by induction on .
Since p is a prime, we have gcd(p 1 , . . . , p k ) = 1 for every decomposition p = p 1 + . . . + p k as above. It follows, using our main theorem, that the action of Σ p1 × · · · × Σ p k on Π p is essentially pointed-free, in the sense that for every subgroup H ⊂ Σ p1 × · · · × Σ p k , the fixed points space Π H p is contractible. Hence the action of G on Π p is essentially pointed-free, which implies that the map |Π p | ∧ hΣp (Σ p /G) + → |Π p | ∧ Σp (Σ p /G) + , which is the same as the map |Π p | / hG → |Π p | / G , is an equivalence.
Corollary 8. 19. If p is an odd prime, then there is a homotopy cofibration sequence Proof. Consider the upper right corner of the square in Proposition 8.18. Since p > 2, we have Corollary 8.20. If p is an odd prime and ≥ 1, then the space |Π p | ∧ Σp S p−1 is p-local.
Proof. By Corollary 8.19 it is enough to prove that the spaces |Π p | ∧ hΣp S −1 and |Π p | ∧ hΣp S p−1 are each p-local. Since ≥ 1 and p > 2 both of these spaces are simply connected. Therefore, it is enough to show that the integral homology of these spaces is p-torsion. We will spell out the proof that H * (|Π p | ∧ hΣp S p−1 , Z) is p-torsion. The proof for the other space is similar. Consider the homomorphisms Here the first homomorphism is the transfer in homology associated with the inclusion Σ p−1 → Σ p , and the second homomorphism is the quotient map. It is well-known that the composed homomorphism is multiplication by p = |Σ p /Σ p−1 |. By Theorem 5.10, |Π p | is Σ p−1 -equivariantly equivalent to (Σ p−1 ) + ∧S p−3 . Therefore |Π p | ∧ hΣp−1 S p−1 S p+p−3 . The reduced homology of this space is Z in dimension p + p − 3 and zero in all other dimensions. On the other hand, by Lemma 8.10 the homology of |Π p | ∧ hΣp S p−1 is all torsion. This implies that the composed homomorphism (8.21) is zero in all dimensions. Multiplication by p is therefore zero on the group H * (|Π p | ∧ hΣp S p−1 , Z). It follows that it is a p-torsion group.
We can finally prove our description of the weight p component of the spaces C Lie (S ): Proof of 8. 17. The case p = 2 is well-known (cf. [JTTW63]). Hence assume that p is an odd prime. We again use the cofibre sequence |Π p | + ∧ Σp S p−1 → S p−1 / Σp → |Π p | ∧ Σp S p−1 . It suffices to show that for odd, the p-localisation of |Π p | + ∧ Σp S p−1 is trivial, and for even, any choice of point in |Π p | induces a p-local equivalence S p−1 → |Π p | + ∧ Σp S p−1 .
Consider the pointed Σ p -space |Π p | + ∧ S p−1 . It is easy to check that the isotropy group of every point in this space (aside from the basepoint) is a non-transitive subgroup of Σ p . It follows that p does not divide the order of any of the isotropy groups. Thus for every isotropy group G, the map BG → * induces an isomorphism in homology with F p -coefficients. Therefore, the map |Π p | + ∧ hΣp S p−1 → |Π p | + ∧ Σp S p−1 is a mod p-homology isomorphism and hence a p-local isomorphism.
The space |Π p | + ∧ hΣp S p−1 is a pointed homotopy colimit of spaces of the form (S p−1 ) hG , where G is an isotropy group of |Π p |. The relevant properties of G are that (1) p does not divide the order of G and (2) G contains a transposition. Suppose that is odd. Then H * (S p−1 ) hG ; F p ∼ = {0}, and so (S p−1 ) hG is p-locally trivial. It follows that |Π p | + ∧ (S p−1 ) hΣp is p-locally trivial. Now suppose is even. The space |Π p | + ∧ S p−1 hΣp is a Thom space of a bundle over |Π p | × Σp EΣ p . Since is even, the bundle is orientable, and therefore the Thom isomorphism holds. Any map * → |Π p | × Σp EΣ p is a mod p homology isomorphism. By the Thom isomorphism theorem, it induces a mod p homology isomorphism of Thom spaces S p−1 → |Π p | + ∧ S p−1 hΣp .
For > 2 the space |Π p | ∧ Σp S p−1 is not equivalent to a wedge of spheres.
For = 2, it is equivalent to S 3 if p = 2 and is contractible if p is odd.
Proof. For p = 2, we have seen that |Π p | ∧ Σp S p−1 Σ RP −1 . For = 2 this space is equivalent to S 3 . For > 2 it is not equivalent to a wedge of spheres since its homology has 2-torsion.
Suppose now that p > 2. By Theorem 9.1, the homology with F p coefficients of |Π p | ∧ Σp S p−1 is generated by symbols {i}, where i is an integer with i ≡ 0 or 1 (mod 2(p − 1)) and 1 < i < (p − 1) . The degree of {i} is + i − 1. It is easy to see that for ≤ 2 there are no integers i satisfying these constraints. It follows that for ≤ 2 the mod p homology of |Π p | ∧ Σp S p−1 is trivial. Since the space is p-local by Corollary 8.20, it follows that it is contractible.
For > 2, there is more than one value of i satisfying the constraints. It follows that the space is not contractible. Since it is p-local, it is not equivalent to a wedge of spheres.
Corollary 8. 23. The quotient space |Π n |/ Σn 1 ×···×Σn k is homotopy equivalent to a wedge of spheres if and only if one of the following holds: gcd(n 1 , . . . , n k ) = 1 or p is a prime, n = 2p or 3p, and gcd(n 1 , . . . , n k ) = p Proof. The case gcd(n 1 , . . . , n k ) = 1 was dealt with in Corollary 8. 3. In the second case, Proposition 8.2 tells us that |Π n |/ Σn 1 ×···×Σn k is equivalent to a wedge sum of spaces of the form S n−3 and |Π p | ∧ Σp S n−p−1 . Since n = 2p or 3p, n − p = p where = 1, 2. By Lemma 8.6 and Corollary 8.22 all these spaces are either equivalent to a sphere or are contractible.
In all other cases, let p be the smallest prime that divides gcd(n 1 , . . . , n k ). Then p divides n and n p > 3. It follows that |Π n |/ Σn 1 ×···×Σn k has a wedge summand equivalent to The last factor is contractible by Lemma 8. 6. A quick calculation shows that |B(4, 4)| = 8 and |B(2, 2)| = 1. We already observed that S 5 / Σ2 ∼ = Σ 3 RP 2 . We conclude that there is an equivalence It follows that the reduced homology of |Π 8 | / Σ4×Σ4 is isomorphic to F 2 in dimension 4, to Z 8 in dimension 5, and is zero otherwise. This is consistent with computer calculations by Donau [Don]. 9. The F p -homology of |Π n | ∧ Σn (S ) ∧n We fix a prime p. We shall use Bredon homology to prove the following result: Theorem 9.1. Let ≥ 1 be a positive integer, assumed to be odd if the prime p is odd. If n is not a power of p, then H * Σ|Π n | ∧ Σn S n , F p is trivial.
If n = p k , then H * Σ|Π p k | ∧ Σ p k S p k , F p has a basis consisting of sequences (i 1 , . . . , i k ), where i 1 , . . . , i k are positive integers satisfying: (1) Each i j is congruent to 0 or 1 modulo 2(p − 1).
Throughout this section, we will writeH * (X) =H * (X, F p ) for the reduced singular homology of a space X with coefficients in F p .
We begin by reviewing the definition and basic properties of Bredon homology. Suppose that ν is a Mackey functor for a finite group G. Let X be a simplicial G-set. The Bredon homology groups H Br * (X; ν) of X with coefficients in ν are defined as the homology groups of the simplicial abelian group obtained by applying ν to X degree-wise. If X is a pointed G-set, then we define the reduced Bredon homology of X as the quotient H If ν * is a graded Mackey functor, then the Bredon homology is bigraded. In this case, we define: The Euler characteristic χ * of a pointed simplicial G-set X with respect to a graded Mackey functor ν is defined as the following sequence of integers: Informally speaking, we take Euler characteristic "in the Bredon direction". We extend this notion in the evident way to bigraded submodules of H Br * (X; ν * ). We will now consider the graded Mackey functor µ * which is defined on Σ n -sets by the formula µ * (S) = H * (S n ∧ Σn S + ).
We observe that if H is a subgroup of Σ n , then µ * (Σ n /H) ∼ = H * (S n / H ).
If W is a general simplicial Σ n -set, then there is a well-known spectral sequence of signature H Br s (W ; µ t ) ⇒ H s+t (S n ∧ Σn W ). We will see below that for W = |Π n | , the spectral sequence collapses at E 2 , and we may therefore focus on calculating the Bredon homology groups H Br * (|Π n | ; µ * ). The next proposition says that the Mackey functor µ * satisfies the hypotheses needed for the main theorem of [ADL16] to apply. Let C G (H) denote the centraliser of the subgroup H in G.
Proposition 9. 3. The Mackey functor µ * (S) = H * (S n ∧ Σn S + ) has the following properties: (1) If Z is a Σ n -set whose cardinality is coprime to p and S is any Σ n -set, then the composed homomorphism µ * (S)

Proof.
The proof is similar to [ADL16,Proposition 11.4]. Since homology is taken with F p coefficients, the Mackey functor µ * takes values in F p -vector spaces. It is well-known that the composite homomorphism µ * (S) tr → µ * (S × Z) → µ * (S) is, in general, multiplication by the cardinality of Z. If the cardinality is coprime to p, then the homomorphism is an isomorphism. This proves (1).
For (2), let σ ∈ ker(C Σn (D) → π 0 C GLn(R) (D)). Then σ is an element of Σ n , σ centralises D, and the linear automorphism of R n induced by σ can be connected to the identity by a path that goes through linear transformations that centralise the action of D. It follows that σ acts on S n by a map that is D-equivariantly homotopic to the identity. Therefore, σ acts on S n / D by a map that is homotopic to the identity. In particular, it acts trivially on µ * (Σ n /D) = H * (S n / D ). (See the second part of the proof of [ADL16, Proposition 11.4] for a very similar argument spelled out in more detail).
Finally, for (3) suppose that p is odd and τ ∈ Σ n is an odd permutation of order 2 centralising D. We need to show that τ acts on H * (S n / D ) as multiplication by (−1). As explained in the last part of the proof of [ADL16,Proposition 11.4], an involution τ on a space W acts by (−1) on the homology of W (with coefficients in a group where 2 is invertible) if and only if (τ − 1) is an isomorphism on homology. This in turn is equivalent to requiring that (τ − 1) induces a quasi-isomorphism on the chain complexC * (W ) of reduced singular chains with coefficients in F p on W . We apply this to the case W = S n / D . There is a natural homotopy equivalence betweeñ C * (S n / D ) and a homotopy colimit of chain complexes of the formC * ((S n ) A ), where A ranges over subgroups of D. It is not difficult to show that τ acts by (−1) on the homology of each fixed point space (S n ) A . It follows that τ − 1 induces a quasi-isomorphism on each chain complexC * ((S n ) A ) and therefore it induces a quasi-isomorphism onC * (S n / D ). This in turn implies that τ acts by (−1) on H * (S n / D ).
By [ADL16, Lemma 3.8 and Theorem 1.1] it follows that H Br s (|Π n | ; µ t ) = 0 unless n = p k for some positive integer k and s = k − 1. Therefore H * (S n ∧ Σn |Π n | ) vanishes unless n is a power of p. In the case n = p k the spectral sequence collapses, and there is an isomorphism H * +k−1 (S p k ∧ Σ p k |Π p k | ) ∼ = H Br k−1 (|Π p k | ; µ * ).
We will therefore focus on calculating the Bredon homology groups H Br k−1 (|Π p k | ; µ * ). Let us describe explicitly the chain complex that we will use.
We define a reduced versionμ * of µ * on pointed sets by settingμ * (S) = H * (S n ∧ Σn S) for S a pointed Σ n -set. To calculate the reduced Bredon homology of |Π p k | , write it as the geometric realisation of a pointed simplicial set with an action of Σ p k (we will describe an explicit model below). Applyingμ * levelwise to |Π p k | yields a graded simplicial abelian group. We refer to the associated normalised chain complex as the Bredon chains on |Π p k | . The bigraded homology groups of this chain complex are precisely the Bredon homology groups H Br * (|Π p k | ; µ * ). Since the Bredon chain complex is concentrated in the single degree (k − 1), its homology is essentially determined by the Euler characteristic in the Bredon direction (cf. Definition 9.2).
To determine the Euler characteristic, we will analyseμ * (Π p k ,i ), where Π p k ,i is the quotient of the set of i-simplices of |Π p k | by the subset of degenerate simplices. We will show thatμ * (Π p k ,i ) splits as a direct sum, and that most of the summands can be arranged in isomorphic pairs in adjacent dimensions, so as to cancel out and contribute nothing to the Euler characteristic.
Finally, we will calculate the Euler characteristic of the complex obtained from summands that are not cancelled out. This approach is essentially the same as was taken in [AM99] towards calculating the homology of the homotopy orbit space S p k ∧ hΣ p k |Π p k | p k . However the situation in the present paper is simpler since we are only calculating the Euler characteristic and therefore we do not need to determine the actual boundary homomorphism between the various summands in our decomposition.
The set Π p k ,i is the wedge sum of sets of the form Σ p k /H + , where H ranges over a set of representatives of stabilisers of nondegenerate i-simplices of |Π p k | . It follows thatμ * (Π p k ,i ) is isomorphic to a corresponding direct sum of groups H * (S p k / H ). Isotropy groups of Π p k ,i are products of wreath products of symmetric groups (see Section 9.2 below for a more precise statement). Therefore, before describing H * (S p k / H ) for a general isotropy group H, we review some results about the homology of X ∧n / Σn for general X. 9.1. Homology of Symmetric Smash Products. Before describing the homology, let us recall the very useful general fact, due to Dold [Dol58], that the homology of a symmetric (smash) product of a pointed space X depends only on homology of X. Dold proved the result for integral homology. We need a version of it for homology with mod p coefficients. A proof can be found in Bousfield's unpublished manuscript [Bou67], where an explicit description of the mod p homology of symmetric products is given. For the reader's convenience, we shall include a proof along the lines of Dold's argument.
Lemma 9. 4. The F p -homology groups of X ∧n / Σn only depend on the homology of the pointed space X. More precisely, there exists an endofunctor G of the category of non-negatively graded F p -vector spaces such that H * (X ∧n / Σn , F p ) is isomorphic to G( H * (X)), naturally in X.
Proof. For a set S, let F Fp (S) denote the F p -vector space with basis S. If S is pointed, letF Fp (S) be the quotient of F Fp (S) by the subspace generated by the basepoint. Note that there is a natural isomorphism of vector spacesF Fp (S ∧n / Σn ) ∼ = (F Fp (S)) ⊗n Σn . Let Sing • (X) be the pointed simplicial set of singular simplices of X. Let C • (X) = F Fp (Sing • (X)) be the simplicial F p -vector space generated by X • and writeC • (X) =F Fp (Sing • (X)) for the reduced F p -chains on X. The reduced homology of X ∧n / Σn is the homology of the simplicial vector spaceC • (X ∧n / Σn ), and this simplicial vector space is isomorphic toC • (X) ⊗n Σn . Let N be the normalised chain complex functor from the category of simplicial F p -vector spaces to the category of non-negative chain complexes over F p . Let Γ be the inverse of N provided by the Dold-Kan correspondence. Then H * (X ∧n / Σn ) is naturally isomorphic to H * N (ΓNC • (X)) ⊗n Σn .
It follows that the functor X → H * (X ∧n / Σn ) from pointed spaces to graded vector spaces factors as the composition of the reduced chain complex functor X → NC • (X) =C * (X), and the functor C → H * N (ΓC) ⊗n Σn from chain complexes to graded vector spaces.
Clearly, this is a homotopy functor. Therefore, it factors through the homotopy category of nonnegative chain complexes over F p . It is well known that this homotopy category is equivalent to the category of graded vector spaces, and H * induces the equivalence of categories. Therefore, H * (X ∧n / Σn ) is naturally isomorphic to H * N (Γ H * (X)) ⊗n It follows that in order to describe the homology of X ∧n / Σn for general X, it is enough to complete this computation for X a wedge of spheres. We will describe the direct sum n≥0 H * (X ∧n / Σn ) as a bigraded vector space.
Definition 9.5. For every integer k ≥ 0, we define an endofunctor F k of the category of graded F p -vector spaces as follows: For k = 0, the functor F 0 is the identity.
(This convention at p = 2 is slightly non-standard, we will explain the reason for it below), (4) We have i j = 1 for all 1 ≤ j ≤ k.
It is easy to check that F k is a well-defined endofunctor of graded vector spaces. Moreover, F k preserves direct sums.
Suppose V is a graded vector space. Then k≥0 F k (V ) is a graded vector space. We endow it with a second grading, which we call "weight": elements of F k (V ) are given weight p k .
For a graded vector space V , define S(V ) to be the free symmetric algebra on V if p is odd, and the exterior algebra on V if p = 2. Any grading of V extends to a grading of S(V ) in the usual way: the degree of a product is the sum of degrees. Let S n (V ) be the direct summand of S(V ) consisting of products of exactly n elements of V . where H * (X ∧n / Σn ) corresponds to elements of weight n.
Remark 9.7. This is not an isomorphism of algebras. Information about the algebra structure on n≥0 H * (X ∧n / Σn ) can be found, for example, in the paper of Milgram [Mil69].
Remark 9.8. For odd primes, our conventions follow closely those of Nakaoka [Nak59]. When p = 2, our conventions differ in that we include in F k elements (i 1 , . . . , i k ; v) satisfying the equality i 1 = i 2 + · · · + i k + |v|. In this case, the element (i 1 , . . . , i k ; v) stands in for the square of the element (i 2 , . . . , i k ; v). We are using the evident isomorphism of graded vector spaces between the polynomial algebra P (x) and the exterior algebra Λ(x 1 , x 2 , x 4 , . . . , x 2 i , . . . , ). This is also why we defined S(V ) to be the exterior rather than polynomial algebra when p = 2. The purpose of this convention is the following: when V is a vector space with one generator (e.g., when V is the reduced homology of a sphere), we want homogeneous summands S n (V ) to be null for n > 1. All this is for the sake of Lemma 9.16 below.
Proof of 9.6. The proposition is well-known. The case when X is a sphere can be read from Nakaoka's [Nak59], with the aforementioned change of conventions at the prime 2. (Nakaoka's paper is written in terms of cohomology, but as far as vector space dimension goes, there is no difference). It is easy to see that both sides of the claimed equation take wedge sums in the variable X to tensor products. It follows that the formula is valid for X a wedge of spheres. By Lemma 9.4, the result holds for a general space X. One can also read off the result from [Bou67] or [Mil69]. Now we can write an explicit formula for H * (X ∧n / Σn ). For this, we first introduce some notation: Definition 9. 9. A p-partition of the integer n is a partition into components of sizes that are powers of the prime p. A p-partition of n will be encoded by a sequence (a 0 , a 1 , . . .) of nonnegative integers (almost all of them zero) such that n = k≥0 a k p k . We write P (n) for the set of p-partitions of n.
The following statement is an immediate consequence of Proposition 9.6. S a k (F k H * (X)). 9. 2. The Homology of Orbit Spaces of Isotropy Groups. Our next task is to describe the vector spaces H * (X ∧n / H ) for H any isotropy group of the suspended partition complex |Π n | and X any pointed space.
We begin by reviewing the structure of isotropy groups of simplices in |Π n | for n > 1. Write P n for the poset of all partitions of {1, . . . , n}. Similarly as in Section 2.9, the space |Π n | arises as the realisation of the following simplicial set: In particular, this means that there is a single non-basepoint zero-simplex. Notice that for every equivalence class of x i , the chain σ induces a chain of partitions of this class of length (i − 1). We call these subchains the restrictions of σ to the equivalence classes of x i .
The symmetric group Σ n permutes such chains of partitions σ. We say that two chains σ, σ are of the same type if they are in the same orbit under this action. More generally, we say that two chains of partitions of two possibly different sets are of the same type if there is a bijection between the sets that takes one of the chains to the other.
Let σ = [0 < x 1 < . . . < x i <1] be an increasing chain of partitions of the set {1, . . . , n} as above. Write K σ ⊂ Σ n for the isotropy group of this chain. Then K σ permutes the equivalence classes of x i (the coarsest partition in the chain). Two equivalence classes of x i are in the same orbit of K σ if and only if the restrictions of σ to these two equivalence classes are of the same type. Let us list the different types of chains that occur among the restrictions of σ to the classes of x i as type 1, type 2, etc. Let K j denote the isotropy group of a chain of type j. Suppose that among the various restrictions of σ to the classes of x i , there are exactly i j many classes of type j.
Then there is an isomorphism

Proof.
We list the different types of chains that occur among restrictions of σ to classes of x i as type 1, type 2, . . . . For each type j, we assume that there are i j many x i -classes of type j, and that each of them has cardinality n j . We again write K j for the isotropy group of any chain of type j.
The discussion preceding Definition 9.12 and Proposition 9.10 together imply For each j, we pick a chain σ j of type j which is obtained by restricting the chain σ to an x i -class. By induction and additivity of each functor F k , we obtain an isomorphism The indexing set of this sum is in fact just E[σ].
Indeed, specifying a p-enhancement Θ of σ up to isomorphism is equivalent to the following data: (1) A p-partition a j k p k = i j for each type j of chains obtained by restricting σ to x i -classes. This corresponds to partitions e i+1 ≥ x i such that there are exactly a j k many e i+1 -classes that contain p k many x i -classes of type j, up to isomorphism. Write b j k,s for the multiplicity of [Θ j k,s ]. Each of these a j k many p-enhancements induces a simultaneous p-enhancement of some p k many x i -classes of type j lying in the same e i+1 -class.
We restrict attention to the summand above corresponding to the isomorphism class of a given p-enhancement Θ = [0 ≤ e 1 ≤ x 1 ≤ . . . ≤ e i ≤ x i ≤ e i+1 ≤1] of σ.
The triples j, k, s indexing the inner tensor product above then corresponds exactly to the possible types t of restrictions of Θ to e i+1 -classes S. Here j encodes the type of the restriction of σ to any x i -class in S, the number k is chosen so that S contains p k many x i -classes, and the index s ∈ {1, . . . , s j k } specifies the type of the restriction of Θ to any of the x i -classes in S. There are b k,s j many e i+1 -classes of this type t, and the claim follows from the second clause of Definition 9.13.
Before coming back to our computation of Bredon homology, we need two rather easy observations.
The second observation covers the case0 = e 1 : In order to examine the pure summands in H * (S n / Kσ ), we introduce the following notation: Definition 9.20. A partition is regular if all its equivalence classes have the same size. More generally, a chain of partitions is regular if each partition in the chain is regular.
Lemma 9.21. A chain of partitions has a unique pure p-enhancement if and only if is a regular chain of partitions of a power of p. Otherwise it has no pure p-enhancements.
Proof. Pure p-enhancements are chains of the form Θ = [0 < e 1 = x 1 < . . . < e i = x i < e i+1 =1] where e j = x j for all 1 ≤ j ≤ i + 1. Therefore, the e j 's are determined by the x j 's and a chain can have at most one pure enhancement.
Suppose Θ is pure. In particular, e i+1 =1. Thus, e i+1 is the indiscrete partition with a single component of size n. By definition of p-enhancements, this implies that the number of equivalence classes of x i is a power of p, and that the restrictions of Θ to the equivalence classes of x i are all pairwise isomorphic. Moreover, the assumption e j = x j for all 1 ≤ j ≤ i implies that the restriction of Θ to any equivalence class of x i is again pure. By induction on the length of the chain, the size of each equivalence class of x i is a power of p, so n must be a power of p as well.
Induction also shows that the chain σ = [0 < x 1 < . . . < x i <1] is regular since all x i are pairwise isomorphic.
On the other hand, it is easy to check that if σ = [0 < x 1 < . . . < x i <1] is a regular chain of partitions of a set of size p k , then setting e j = x j for all j defines a pure p-enhancement of σ.
In particular, if n is not a power of p, then there are no pure summands in the homology of H * (S n / Kσ ), where K σ is any isotropy group of |Π n | . We obtain the following (superfluous) corollary. Of course, a much stronger statement is true: if n is not a power of p, then all the Bredon homology groups vanish by [ADL16] and Proposition 9.3 (cf. the remark following Proposition 9.3).
It remains to calculate the Euler characteristic when n = p k .
Proof of Theorem 9.1. Let n = p k be a power of p. By Proposition 9.19, it suffices to calculate the alternating sum (with respect to the Bredon grading) of the dimensions of the pure summands and check that it matches the dimensions asserted in the statement of the theorem.
By Lemma 9.21 there is a unique pure summand for each isomorphism type of regular chains of partitions of {1, . . . , p k }. Such chains are in bijective correspondence with ordered partitions of k, or equivalently with subsets of {1, . . . , k − 1}.
We will now make the pure summands more explicit. Given an ordered partition (k 1 , . . . , k r ) of k, we pick a regular chain of partitions σ (k1,. ..,kr) representing the corresponding isomorphism class. There is a unique pure p-enhancement Θ of σ, and the corresponding pure summand [Θ]( H * (S )) inside H * (S n / Kσ ) (sitting in Bredon degree r−1) is readily seen to be given by F k1 . . . F kr ( H * (S )). The Bredon complex can therefore be depicted as: