The Free Loop Space Homology of $(n-1)$-connected $2n$-manifolds

Our goal in this paper is to compute the integral free loop space homology of $(n-1)$-connected $2n$-manifolds $M$, $n\geq 2$. We do this when $n\neq 2,4,8$, or when $n\neq 2$ and $\tilde H^*(M)$ has trivial cup product squares, though the techniques used here should extend to a much wider range of manifolds. We also give partial information concerning the action of the Batalin-Vilkovisky operator.

be the free tensor algebra generated by V , and I be the two-sided ideal of the tensor algebra T (V ) generated by the following degree 2n − 2 element where [x, y] = xy − (−1) x y yx denotes the graded Lie bracket in T (V ). Take the quotient algebra U = T (V ) I and the degree −1 maps of graded R-modules d∶ A ⊗ U → U and d ′ ∶ K ⊗ U → A ⊗ U , which are given for any y ∈ U by the formulas If we apply the Jacobi identity to the summands c ij (a j ⊗ [u i , y]) in d ○ d ′ (y) for i < j (keeping in mind that c ij = (−1) n c ji , [u i , [u i , y]] = [u 2 i , y], and that products with χ are identified with zero in U ), we see that Im d ′ ⊆ ker d, so we obtain a chain complex Now take the homology of this chain complex. That is, take the following graded R-modules: One can think of W by first taking the R-submodule W ′ of Σ −1 A ⊗ T (V ) ≅ T (V ) generated by elements that are invariant modulo I under graded cyclic permutations, that is, invariant after projecting to U . Then W is the projection of ΣW ′ onto (A ⊗ U ) Im d ′ .
Our main result says that the homology of this chain complex is the integral free loop space homology of M under some dimension assumptions: The restriction away from 2,4, and 8 traces back to an argument that we use to determine H * (ΩM ), which does not apply to situation where there are cup product squares equal to the fundamental class [M ], or −[M ]. The failure of a degree placement argument to compute certain differentials is another reason that we restrict away from n = 2.
We determine the action of the BV-operator on H * (LM ; Q), in a sense, up-to-abelianization of U when n > 3 is odd. Consider the graded abelianization map T (V ) η → S(V ), where S(V ) is the free graded symmetric algebra generated by V . Since η(χ) = 0, η factors through U η → S(V ). Also, consider the maps A ⊗ U ½ A ⊗η → A ⊗ S(V ) and K ⊗ U ½ K ⊗η → K ⊗ S(V ). Since (½ A ⊗ η) ○ d ′ = 0 and η ○ d = 0, then η and these two maps induce abelianization maps

Fibrations with H-space Fibers
We begin with a few preliminaries that will be used in the proof of Theorem 1.1. Take a fibration be the homology Serre spectral sequence for fibration f . The fibration f is said to be principal if F is a homotopy associative H-space, and if there is a left action F × X → X fitting into a homotopy commutative square: denotes the H-space multiplication on F ). A result of Moore [14] tells us that E inherits the the structure of a left H * (F )-module, which means there is a left action reducing to the Pontrjagin multiplication on E 2 0, * ≅ H * (F ), and differentials respect this action. Thus much of the effort in computing differentials is reduced to determining those emanating from the degree 0 horizontal line in the spectral sequence.
For a more generic fibration f , the induced homotopy fibration sequence is in all cases principal. The H-space structure on ΩB is taken as the one defined by composing loops, and the action ΩB × F θ → F here is defined by applying the homotopy lifting property to loops in B. Since fibrations are characterized by satisfying the homotopy lifting property, in some rough sense we might expect any information gained from the left-H * (ΩB) structure on H * (F ) to have some bearing on the spectral sequence for f . This was exploited by McCleary in [12], using a result of Brown [3] and Shih [16], to give a computation of the free loop space homology of certain low rank Stiefel manifolds. The aim of the following proposition is to strengthen the technique used therein under the condition that F is a homotopy associative H-space, the gain here being that one can to do away with an assumption about certain elements being trangressive. We let E = {E r , d r } denote the homology Serre spectral sequence for the path-loop fibration sequence ΩB Proposition 2.1. Suppose H * (B), H * (ΩB) are torsion free, and F is a homotopy associative H-space. Given z ∈ H * (B), suppose d s (z ⊗ 1) = 0 ∈ E s * , * for 2 ≤ s < r, and z ⊗ y ∈ E 2 * , * survives to E r * , * . Then for every y ∈ H * (F ) and 2 ≤ s < r, δ s (z ⊗ y) = 0 ∈ E s * , * and Here, for clarity, we use⊗ to indicate tensors in H * (ΩB × F ) ≅ H * (ΩB) ⊗ H * (F ).
Proof. First recall the following well-known property (which is essentially the homotopy lifting prop- Take the inclusion φ∶ PB × F → P ev0,f given by φ(ω, a) = (ω, a), and take the the compositē Let the fibration sequence be the product of the path-loop fibration sequence ΩB ⊂ → PB ev1 → B and the trivial fibration sequence F ½ → F * → * . Both the path-loop fibration and the trivial fibration are principal, and ΩB × F is a homotopy associative H-space, since both F and ΩB are. Then this product fibration is a principal fibration sequence, as is apparent in the following commutative diagram Here T is the transposition map, the bottom square is the product of the squares that commute due to the trivial fibration and path-loop fibration being principal. The right vertical composite defines the action ψ of ΩB × F on PB × F , where ψ 0 is the action for the path-loop fibration, given by composing a based loop with a based path at the basepoint.
Consider the commutative diagram of fibration sequences We letÊ = {Ê r ,d r } be the homology Serre spectral sequence for the fibration sequence (1), and γ∶ E →Ê the morphism of spectral sequences induced by diagram (2). One can also easily check that the following diagram of fibration sequences commutes: (3) with the action θ constructed as the restriction ofθ to the subspace ΩB × F , which is the reason for the left-most commutative square. Let ζ∶Ê → E be the morphism of spectral sequences induced by this diagram. The element z ⊗ (1⊗y) ∈Ê 2 * , * survives toÊ r * , * as follows. Inductively, assume that it has survived toÊ s * , * for some 2 ≤ s < r. Since our assumption is that z ⊗ y ∈ E 2 * , * survives to E r * , * , z ⊗ y is not in the image of any differential δ s for 2 ≤ s < r. Since z ⊗ y = ζ s (z ⊗ (1⊗y)), by naturality z ⊗ (1⊗y) is also not in the image of any differentiald s . Now using the fact that the bottom fibration sequence in diagram (2) is principal, and that d s which implies z ⊗ (1⊗y) survives toÊ s+1 * , * . This completes the induction. Finally, inÊ r * , * we have the formulâ which we can use to obtain the following Similarly, δ s (z ⊗ y) = 0 for 2 ≤ s < r.
Proposition 2.1 still manages to hold if F is not an H-space, as long as there is a map G f → F with G an H-space, and we restrict y ∈ Im f * ⊆ H * (F ) in the statement of the proposition. One replaces the fibration sequence (1) in the proof with ΩB × G ⊂×½ → PB × G ev1× * → B × * , and composes it with diagram (3) using the map f . Generally, if F is not an H-space, the proposition holds when z is transgressive, which is the result due to Brown and Shih.
We now turn our focus towards the free loop space fibration sequence The map ϑ is the canonical inclusion ΩB ⊆ LB, and ev 1 is the evaluation map ev 1 (ω) = ω(1). The homology Serre spectral sequence for this fibration sequence will be denoted by and as before E = {E r , d r } is the homology Serre spectral sequence for the path-loop fibration sequence of B. Some basic properties of the free loop space fibration are as follows. The map LB where the multiplication on H * (ΩB) is the Pontrjagin multiplication induced by loop composition on ΩB. The proof of these facts can be found in [12] (for example).
Combining these properties with Propositions 2.1, we gain the following description of the differentials in the spectral sequence E.
such that each v i is primitive. Then for every y ∈ H * (ΩB) and 2 ≤ s < r, we have δ s (z ⊗ y) = 0 ∈ E 2 * , * , and ◻ While it may not be apparent at first, there are instances where this formula will fail to give us enough information to determine some of the higher differentials. For example, if one found themselves in the situation where δ s (z ⊗ y) = 0 for s ≤ r and d r (z ⊗ y) ≠ 0, then z ⊗ y ∈ E r * , * survives to the E r+1 page, while z ⊗ y is not an element in E r+1 * , * . In such case δ s (z ⊗ y) remains mysterious when s > r. An example where this situation happens in practice is the omitted 4-manifold case in Theorem 1.1.

Based Loop Space Homology
Returning to our manifold M in the introduction, in this section we consider the Hopf algebra H * (ΩM ). This is the last piece in the puzzle required to prove Theorem 1.1. By Poincaré duality the only nonzero reduced homology groups of M are in degrees n and 2n. This implies M has a cell decomposition given by attaching an n-cell to an m-fold wedge of n-spheres ⋁ m S n ≃ M − * .
Very generally, if a space Y is formed by attaching a k-cell to a space X via an attaching map S k−1 α → X, and α ′ is its adjoint, the composite with the looped inclusion S k−2 α ′ → ΩX Ωi → ΩY is nullhomotopic, so one obtains a factorization of Hopf algebras through Hopf algebra maps where I is the two-sided ideal generated by α ′ ([S k−2 ]) ∈ H k−2 (ΩX; R). The problem of determining the conditions under which θ is a Hopf algebra isomorphism is part of what is known as the cellattachment problem. One of these, the inert condition, states somewhat suprisingly that θ is a Hopf algebra isomorphism when R is a field if and only if (Ωi) * is a surjection ( [11,8,6]). Here we select k = 2n, Y ≃ M , and X ≃ M − * , and use the inert condition to prove the following: Proof of part (i). In [2], ΩM is shown to be a homotopy retract of Ω(M − * ) when n ≠ 2, 4, 8. Therefore (Ωi) * is a split epimorphism, so we obtain an isomorphism H * (ΩM ; F ) ≅ H * (Ω(M − * ); F ) I for any field F . Moreover, since M − * is homotopy equivalent to ⋁ m S n , the Z-module H * (Ω(M − * ); Z) ≅ T (V ) is torsion-free. Therefore H * (ΩM ; Z) is torsion-free, and the Hopf algebra isomorphism holds for R = Z as well.
Proof of part (ii). We will write u j = (Ωi) * (u j ) ∈ H n−1 (ΩM ), and take u j to be the transgression of a j ∈ H n (M ).
Since the elements u 1 , . . . , u m in H n−1 (Ω(M − * )) are primitive, and there are no monomials of length greater than 2 in degree 2n − 2, the elements u 2 i and [u j , u i ] form a basis for the primitives in H 2n−2 (Ω(M − * )).
Therefore E 2n 0, * ≅ Q, E ∞ n, * ≅ E 2n n, * ≅ W, and E 2n 2n, * ≅ Z, while all other entries in the spectral sequence are zero. But since the nonzero elements in Q and Z are concentrated in degrees k(n − 1) and 2n + k(n − 1) respectively, the differentials δ 2n are zero for degree placement reasons whenever n > 2. Thus these isomorphisms carry over to the infinity page, that is, Generally, one has torsion here when R = Z (or at least in Q, and possibly W), so we must consider a potential extension problem. Once again placement reasons allow us to skirt around the issue.
From the construction of the homology Serre spectral sequence there are increasing filtrations Since the nonzero elements in Q, W, and Z are in degrees k(n−1), n+k(n−1), and 2n+k(n−1), Q, W, and Z pairwise have no nonzero elements in the same degrees when n > 3. Since F n−1, * = F 0, * = Q, we have F n−1,n+k(n−1) = {0}, and we see that F n, * ≅ F 0, * ⊕ E ∞ n, * ≅ Q ⊕ W. Then F 2n−1,2n+k(n−1) = F n,2n+k(n−1) = {0}, so F 2n, * ≅ F n, * ⊕ E ∞ 2n, * , and we have E ∞ 2n, * ≅ F 2n, * = H * (LM ) whenever n > 3. When n = 3, the common nonzero degrees shared between any pair of these three modules are of the form 2(k + 3), and these are only between Q and Z. But since Z is torsion-free and Q = F 0, * is at the bottom filtration, there are no extension issues here either.

Eilenberg-Maclane Spaces and the BV-operator
We will need some information about the action of the BV-operator on products of Eilenberg-Maclane spaces in the proof Theorem 1.2. The approach we take here is similar to the one taken by Hepworth in [10] to compute the BV-operator for Lie groups, which we begin this section with by recalling. Fix R to be a principal ideal domain, and X (homotopy type of a CW -complex) a path-connected topological group with multiplication X ×X µ → X. This makes LX into topological group with multiplication LX × LX Lµ → LX defined by point-wise multiplication of loops (ω ⋅ γ)(t) = ω(t) ⋅ γ(t). There is a well-known homeomorphism These homeomorphisms are equivariant with respect to our action S 1 × LX ν → LX, and the action ν∶ S 1 × X × ΩX → X × ΩX defined by the formulā where ω t (s) = ν(t, ω)(s) = ω(s + t). On homology we have a commutative square where∆(e) =ν * ([S 1 ] ⊗ e). Clearly, after transposing X and S 1 ,ν is the composite with ev∶ S 1 × ΩX → X the evaluation map ev(t, ω) = ω(t) = ω t (0), and φ∶ S 1 × ΩX → ΩX defined by φ(t, ω) = ω t (0) −1 ⋅ ω t . Thus, if H * (ΩX; R) is a free R-module, so that (for simplicity) the cross product H * (X; R) ⊗ H * (ΩX; R) × → H * (X × ΩX; R) is an isomorphism, and the coproduct on an element b ∈ H * (ΩX; R) has the form △ where ǫ∶ H * (ΩX; R) → R is the augmentation. To complete this formula one needs to determine the maps φ * and ev * . This latter map defines the homology suspension σ∶ H * (ΩX; R) → H * +1 (X; R), σ(a) = ev * ([S 1 ] ⊗ a), which satisfies the formula (9) σ(ab) = σ(a)ǫ(b) + ǫ(a)σ(b) for any product ab ∈ H * (ΩX; R) induced by the loop composition multiplication on ΩX. In particular, σ is zero on decomposable elements. One can derive this formula by observing that the following diagram commutes and that point-wise multiplication of based loops Ωµ on ΩX is homotopy commutative and homotopic to the loop composition multiplication on ΩX (this is a mapping space analogue of Theorem 5.21, Chapter III in [19]). Alternatively, it is a consequence of the Homology Suspension Theorem ( [19], Chapter VIII). The map κ(a) = φ * ([S 1 ]⊗a) is a bit more mysterious. At the very least, when µ is commutative one obtains an analogous commutative diagram for φ together with a derivation formula κ(ab) = κ(a)b + aκ(b), while for the case of compact Lie groups, κ is trivial since H * (ΩX) is concentrated even degrees. We consider the case where X is an Eilenberg-Maclane space K(R, n). These can be taken to be commutative topological groups, and we may write K(G, n−1) = ΩK(G, n) with commutative multiplication induced by the one on K(R, n), which by the way is homotopic to the loop composition multiplication.
Next, recall the suspension isomorphism H n−1 (K(R, n − 1); R) ≅ → H n (ΣK(R, n − 1); R), sending a ↦ [S 1 ] ⊗ a, factors as the composite where the last map is the adjoint isomorphism. Since the evaluation map S 1 ×K(R, n−1) ev → K(R, n) restricts to the constant map on both the left and right factors, it factors as the composite where the last map ev ′ (also known as the evaluation map in the literature) is the adjoint of the identity map K(R, n−1) ½ → K(R, n−1). Since the identity is a classifying map of ι n−1 , by the above factorization of the suspension, its adjoint ev ′ is a classifying map of [S 1 ] ⊗ ι n−1 . The proposition now follows using equation (8).
The BV-operator has a very clean form on decomposable elements when we take our multiplication on H * (LX) to be the one induced by point-wise multiplication of loops Lµ (instead of the multiplication (Ωµ × µ) ○ (½ × T × ½) based on each coordinate of ΩX × X ≅ LX). Tamanoi [17] gave a derivation formula with respect to this product which is a straightforward consequence of the following commutative diagram Both multiplications on LX are equal when the multiplication on X is commutative. Since this is the case for K(R, n), our formula in Proposition 5.1 satisfies The derivation formula can also be used to determine how the BV-operator interacts with the crossproduct, as we see in the following: Proposition 5.2. Let X = X 1 × ⋯ × X k be a product of topological groups (X i , µ i ). Then the BV-operator for LX ≅ LX 1 × ⋯ × LX k satisfies j=1 a j and k 1 = 0.
Proof. It suffices to prove the statement for length-2 products X = X 1 × X 2 . One can then iterate to obtain the general formula. Since the inclusion of the left factor LX 1 ½× * → LX 1 × LX 2 induces the map on homology sending a ↦ a × 1 for any a, by naturality of the BV-operator we have ∆(a 1 × 1) = (½ × * ) * (∆(a 1 )) = ∆(a 1 ) × 1. Similarly, ∆(1 × a 2 ) = 1 × ∆(a 2 ). Since X is a topological group with multiplication µ defined by the composite X × X Therefore (a 1 × 1)(1 × a 2 ) = a 1 × a 2 with respect to this induced product, and by the derivation formula we have We have, for the sake of simplicity, been restricting X to be a topological group. Some of the material above however extends (up-to-homotopy) to where X is a homotopy associative H-space. In this scenario h is a homotopy equivalence since it defines is a weak equivalence between the free loop fibration and the trivial fibration. If X has an inverse −½∶ X → X, x ↦ x −1 , the null homotopy H∶ X × X × I → X, with H 0 = * and H 1 = ½ × −½, allows us to define the homotopy inverse h −1 just as before, except this time composing the loop ω(0) −1 ⋅ ω with the based path given by H t (ω(0) −1 , ω(0)), and the actionν will have a similar form.
In the case of rational coefficients, a simply connected H-space X has a rational decomposition X Q ≃ ∏ i K(Q, n i ), and the classifying maps ΣK(Q, n i − 1) → K(Q, n i ) can be identified with the Freudenthal suspension S ni and φ ∞ restricts to the maps Q ηq → Q[W ] ≅ E ∞ 0, * , W ηw → W ⊗ Q[W ] ≅ E ∞ n, * , and Z ηz → Q{β} ⊗ Q[W ] ⊆ E ∞ 2n, * (note W ≅ V , Q{β} ≅ K, and Q[W ] ≅ S(V ) in the introduction). Let F be the filtration of H * (LP ; Q) associated with the spectral sequence E. Notice E ∞ n, * ≅ F n,n+ * Q[W ], and Q[W ] is concentrated in degrees k(n − 1), while W is concentrated in degrees n + k(n − 1), which are never equal when n > 3, so they do not share any nonzero elements in the same degree. Similarly, E ∞ 2n, * ≅ F 2n,2n+ * F n,2n+ * , F n, * ≅ Q[W ] ⊕ (W ⊗ Q[W ]) is concentrated in degrees k(n − 1) and n + k(n − 1), and Z is concentrated in degrees 2n + k(n − 1), which are never equal when n > 3. Therefore, with respect to our isomorphism H * (LM ; Q) ≅ Q ⊕ W ⊕ Z, (Lf ) * restricts to the maps η q , η w , and η z on each summand.

Acknowledgements
The second author was supported by a Leibniz-Fellowship from Mathematisches-Forschungsinsitut-Oberwolfach and an Invitation to the Max-Planck-Institut für Mathematik in Bonn. Both authors are grateful to the MFO's hospitality to let them spend some time together to work on this project, and Professor Jie Wu and John McCleary for their helpful comments.