$\mathcal{E}_\infty$ ring spectra and elements of Hopf invariant $1$

The $2$-primary Hopf invariant $1$ elements in the stable homotopy groups of spheres form the most accessible family of elements. In this paper we explore some properties of the $\mathcal{E}_\infty$ ring spectra obtained from certain iterated mapping cones by applying the free algebra functor. In fact, these are equivalent to Thom spectra over infinite loop spaces related to the classifying spaces $B\mathrm{SO},\,B\mathrm{Spin},\,B\mathrm{String}$. We show that the homology of these Thom spectra are all extended comodule algebras of the form $\mathcal{A}_*\square_{\mathcal{A}(r)_*}P_*$ over the dual Steenrod algebra $\mathcal{A}_*$ with $\mathcal{A}_*\square_{\mathcal{A}(r)_*}\mathbb{F}_2$ as an algebra retract. This suggests that these spectra might be wedges of module spectra over the ring spectra $H\mathbb{Z}$, $k\mathrm{O}$ or $\mathrm{tmf}$, however apart from the first case, we have no concrete results on this.


Introduction
The 2-primary Hopf invariant 1 elements in the stable homotopy groups of spheres form the most accessible family of elements. In this paper we explore some properties of the E ∞ ring spectra obtained from certain iterated mapping cones by applying the free algebra functor. In fact, these are equivalent to Thom spectra over infinite loop spaces related to the classifying spaces BSO, BSpin, BString.
We show that the homology of these Thom spectra are all extended comodule algebras of the form A * A(r) * P * over the dual Steenrod algebra A * with A * A(r) * F 2 as an algebra retract. This suggests that these spectra might be wedges of module spectra over the ring spectra HZ, kO or tmf, however apart from the first case, we have no concrete results on this.
Our results and methods of proof owe much to work of Arunas Liulevicius [13,14] and David Pengelley [18][19][20], and are also related to work of Tony Bahri and Mark Mahowald [2]. However we use some additional ingredients: in particular we make use of formulae for the interaction between the A * -coaction and the Dyer-Lashof operations in the homology of an E ∞ ring spectrum described in [6]. We also take a slightly different approach to identifying when the homology of a ring spectrum is a cotensor product of the dual Steenrod algebra A * over a finite quotient Hopf algebra A(n) * , making use the fact that the dual Steenrod algebra is an extended A(n) * -comodule; in turn this is a consequence of the P -algebra property of the Steenrod algebra A * .
Conventions: We will work 2-locally throughout this paper, thus all simply connected spaces and spectra will be assumed localised at the prime 2, and M S will denote the the category of S-modules where S is the 2-local sphere spectrum as considered in [10]. We will write S 0 for a chosen cofibrant replacement for the S-module S and S n = Σ n S 0 . When discussing CW skeleta of a space X we will always assume that we have chosen minimal CW models in the sense of [8] so that cells correspond to a basis of H * (X) = H * (X; F 2 ). Notation: When working with cell complexes (of spaces or spectra) we will often indicate the mapping cone of a coextension g of a map g : S n → S k by writing X ∪ f e k ∪ g e n+1 . ( ( P P P P P P P P P P P P P ΣX X Of course this notation is ambiguous, but nevertheless suggestive. When working stably with spectra we will often write h : S n+r → S k+r for the suspension Σ r h of a map h : S n → S k . We will also often identify stable homotopy classes with representing elements.
Proposition 1.1. The following CW spectra exist: Sketch of proof. In each of the iterated mapping cones below, we will denote the homology generator corresponding to the unique cell in dimension n by x n . The case of S 0 ∪ η ∪ 2 e 3 is obvious.
Consider the mapping cone of ν, C ν = S 0 ∪ ν e 4 . As νη = 0, there is a factorisation of η on the 4-sphere through C ν .
Thus these Thom spectra are examples of 'iterated Thom complexes' similar in spirit to those discussed in [7].
Each skeletal inclusion factors uniquely through an infinite loop map j r , where Q = Ω ∞ Σ ∞ is the free infinite loop space functor. We can also form the associated Thom spectrum M j r which is an E ∞ ring spectrum admitting an E ∞ morphism M j r → M O 2 r factoring the corresponding skeletal inclusion.
The first two are induced from well known orientations, while the third relies on unpublished work of Ando, Hopkins & Rezk [1]. Actually such morphisms can be produced using the reduced free commutative S-algebra functor P of [5], which has a universal property analogous to that of the usual free functor P of [10].
Proof. The existence of such morphisms depends on the universal property of P. The proof that those of the first kind are equivalences depends on a comparison of the homology rings using Theorem 2.3 below.
The homology of M j r can be determined from that of the underlying infinite loop space using the Thom isomorphism, while that for the others depends on a general description of the homology of H * ( PX) which can be found in [5].
Theorem 2.3. The homology rings of the Thom spectra M j r are given by The homomorphisms H * (M j r ) → A * induced by the E ∞ orientations M j r → HF 2 have as images

Some coalgebra
First we recall a standard algebraic result, for example see [19, lemma 3.1]. We work over a field k and with k-vector spaces. We will set ⊗ = ⊗ k . There are slight modifications required for the graded case which we leave the reader to formulate, however as we work exclusively in characteristic 2, these have no significant effect in this paper. We refer to the classic paper of Milnor and Moore [16] for background material on coalgebra.
Let A be a commutative Hopf algebra over a field k, and let B be a quotient Hopf algebra of A. We denote the product and antipode on A by ϕ A and χ, and the coaction on a left comodule D by ψ D . We will identify the cotensor product Here the codomain has the diagonal A-comodule structure, while the domain has the left A-comodule structure.
Here is an easily proved generalisation of this result.
Lemma 3.2. Let C be a commutative B-comodule algebra and let D be a commutative Acomodule algebra, then there is an isomorphism of A-comodule algebras where the domain has the diagonal left A-coaction and C ⊗ D has the diagonal left B-coaction.
Explicitly, on an element the isomorphism has the effect Similarly the inverse is given by . This dualises as follows: If L is a quotient Hopf algebra of H, then H is an extended left or right L-comodule, i.e., More generally, according to Margolis [15, pps 193 & 240], if H is a P -algebra then a result of the first kind holds for any finite dimensional sub-Hopf algebra K.
We need to make use of the finite dual of a Hopf algebra H, namely Then H o becomes a Hopf algebra with product and coproduct obtained from the adjoints of the coproduct and product of H. We will say that H is a P -coalgebra if H o is a P -algebra.
These are isomorphisms of left or right A-comodules for suitable comodule structures on the right hand sides.
To understand the relevant A-comodule structure on (A B k) ⊗ M , note that there is an isomorphism of left A-comodules where the right hand factor is the isomorphism of Lemma 3.3.
Crucially for our purposes, for a prime p, the Steenrod algebra A * is a P -algebra in the sense of Margolis [15], i.e., it is a union of finite sub-Hopf algebras. When p = 2, and it follows from the preceding results that if n 0, A * is free as a right or left A(n) * -module, see [15, pps 193 & 240]. Dually, (A * ) o = A * and A * is an extended A(n) * -comodule: Given this, we see that for any left A(n) * -comodule M * , as vector spaces In fact this is also an isomorphism left A * -comodules.
Here is an explicit description of isomorphisms of the type given by Lemma 3.3. For n 0, we will use the function For an natural number r, write where 0 r ′′ (n, i) < e n (i). We note that , . . .).
We will indicate elements of A(n) * by writing z for the coset of z which is always chosen to be a sum of monomials ζ s 1 1 ζ s 2 2 · · · ζ s ℓ ℓ with exponents satisfying 0 s i < e n (i).
Proposition 3.5. For n 0 there is an isomorphism of right A(n) * -comodules given on basic tensors by We will also use the following result to construct algebraic maps in lieu of geometric ones. The proof is a straightforward generalisation of a standard one for the case where B = k.
As an example of the multiplicative version of this result, suppose that M is an A-comodule algebra which is augmented. Then there is a composite homomorphism of B-comodule algebras α : M → k → N giving rise homomorphism of A-comodule algebras 4. The homology of M j r for r = 1, 2, 3 Now we analyse the the specific cases for H * (M j r ) for r = 1, 2, 3. Since some of the details differ in each case we treat these separately. In each case there is a commutative diagram of commutative A * -comodule algebras (4.1) where the left A * -coaction is determined by To calculate the coaction on the other generators Q I x 2 and Q J x 3 we follow [6] and use the right coaction where ψ(z) = i α i ⊗ z i and χ is the antipode of A * . So In general, if z has degree m, then By (4.3), We also have Combining these we obtain 4) or equivalently, We will consider the sequence of elements X 1,1 and X 1,s ∈ H 2 s −1 (M j 1 ) (s 2) defined by We claim the X 1,s have the following right and left coactions: ψX 1,s = 1 ⊗ X 1,s + ζ 1 ⊗ X 2 1,s−1 + · · · + ζ s−3 ⊗ X 2 s−3 1,3 + ζ s−2 ⊗ X 2 s−2 1,2 + ζ s−1 ⊗ X 2 s−2 1,1 + ζ s ⊗ 1. (4.7) To prove these, we use induction on s, where the early cases s = 1, 2, 3 are known already. For the inductive step, assume that (4.6) holds for some s 3. Then giving the result for s + 1.
This shows that the restriction of ρ to the subalgebra generated by the X 1,s is an isomorphism of A * -comodule algebras In the algebra H * (M j 1 ), the regular sequence X 1,s (s 1) generates an ideal This is not an A * -subcomodule since for example, However under the induced A(0) * -coaction the last term becomes trivial, in fact ψ ′ X 1,3 = 1 ⊗ X 1,3 + ζ 1 ⊗ X 2 1,2 , where we identify elements of A(0) * with representatives in A * . More generally, by (4.7), for s 2, ψ ′ X 1,s = 1 ⊗ X 1,s + ζ 1 ⊗ X 2 1,s−1 . It follows that I 1 is an A(0) * -invariant ideal.
We have the following splitting result.

The homology of M j 2 . We have
with right coaction satisfying Furthermore, so the left A(1) * -coproduct We also have Now we define a sequence of elements X 2,s (s 1) by x 7 if s = 3, A inductive calculation shows that for s 4, So this sequence is regular and generates an A(1) * -invariant ideal The next result follows using similar arguments to those in the proof of Proposition 4.1 using the diagram (4.1).

Proposition 4.4. There is an isomorphism of A * -comodule algebras
. We have the following splitting result analogous to Proposition 4.3.

4.3.
The homology of M j 3 . In H * (M j 3 ), consider the regular sequence We leave the reader to verify that the ideal is A(2) * -invariant. The proof of the following result is similar to those of Propositions 4.1 and 4.4 using the diagram (4.1).

Proposition 4.6. There is an isomorphism of A * -comodule algebras
We have the following splitting result analogous to Propositions 4.3 and 4.5.
Proposition 4.7. There is a splitting of A * -comodule algebras

Some other examples
The approach we have used to proving algebraic splittings of the homology of E ∞ Thom spectra can be be used to rederive 5.1. An example related to kU. Our first example is based on similar ideas to those used to construct the spectra M j r , but using Spin c . The low dimensional homology of BSpin c can be read off from Theorem A.2 and Remark A.3. Passing to the Thom spectrum over the 7-skeleton (BSpin c ) [7] we have for its homology 3,0 , a 7,0 }. For our purposes, the fact that there are two 4-cells is problematic, so we instead restrict to a smaller complex. The map K(Z, 2) [2] × BSpin [7] → BSpin c induces an epimorphism in cohomology, and the resulting map S 2 ∨ BSpin [7] → BSpin c gives a monomorphism in homology with image 3,0 , a 7,0 }.
The Thom spectrum over this space has the cell structure of the form (S 0 ∪ η e 2 ) ∪ ν e 4 ∪ η e 6 ∪ 2 e 7 . We can factor the skeletal inclusion through an infinite loop map and obtain an E ∞ Thom spectrum M j c over Q(S 2 ∨ BSpin [7] ). The homology of this is It is easy to see that there is a morphism of E ∞ ring spectra P(S 0 ∪ η e 4 ∪ η e 6 ∪ 2 e 7 ) → kU inducing an epimorphism on H * (−) under which Now we define a sequence of elements X s in H * (M j c //w) as follows: This forms a regular sequence and the induced coaction over the quotient Hopf algebra Recall that We have proved the following analogues of earlier results.

Some speculation
Our algebraic splittings of H * (M j r ) are consistent with spectrum-level splittings. Indeed, in the case of r = 1, a result of Mark Steinberger [9] already shows that M j 1 splits as a wedge of suspensions of HZ and HZ/2 s for s 1, all of which are HZ-module spectra.
Using Lemma 3.2, it is easy to see that if a spectrum X is a module spectrum over one of HZ, kO or tmf then its homology is a retract of the extended comodule A * A(r) * H * (X) for the relevant value of r; a similar observation holds for a module spectrum over kU and A * E(1,2) * H * (X). Thus our algebraic results provide evidence for the following conjectural splittings. Conjecture 6.1. As a spectrum, M j 2 is a wedge of kO-module spectra, M j 3 is a wedge of tmf-module spectra and M j c is a wedge of kU-module spectra.
Here the phrase 'module spectra' can be interpreted either purely homotopically, or strictly in the sense of [10].
Related to this conjecture, and indeed implied by it, is the following where we know that analogues hold for the cases M j 1 , M j 2 , M j c , i.e., the natural homomorphisms are epimorphisms.
We already know this is true up to degree 16, and also holds rationally.
Appendix A. The homology of connective covers of BO We review the structure of the homology Hopf algebras H * (BO n ) = H * (BO n ; F 2 ) for n = 1, 2, 4, 8. The dual cohomology rings were originally determined by Stong but later a body of literature due to Bahri, Kochman, Pengelley as well as the present author evolved describing these homology rings. We will use the Husemoller-Witt decompositions of [3] to give explicit algebra generators; the actions of Steenrod and Dyer-Lashof operations on these can be determined using work of Kochman and Lance [11,12].
We recall that there are polynomial generators a k,s ∈ H 2 s k (BO) (k odd, s 0) such that is a polynomial sub-Hopf algebra and For each h 1, there is a Hopf algebra monomorphism We may identify H * (M O n ) with H * (BO n ) using the Thom isomorphism which is an isomorphism of algberas over the Dyer-Lashof algebra but not over the Steenrod algebra. To avoid excessive notation we will often treat the Thom isomorphism as an equality and write a For completeness, we also describe the homology of BSpin c in this algebraic form since we are not aware of this being documented anywhere. , where w k ∈ H k (BSO) is the image of the k-th Stiefel-Whitney class, x ∈ H 2 (K(Z, 2)) and x 2 t ∈ H 2 t+1 (K(Z, 2)) transgresses to d 2 t+1 +1 (x 2 t ) = w 2 t +1 (mod decomposables). Lemma B.1. Let s 1. If k ∈ N, then Q k ζ s ∈ I(s − 1); more generally, for r 0, Q k (ζ 2 r s ) ∈ I(s + r − 1).
Proof. We make use of the results of [6, section 5].