Homology $H_*(TOP, \mathbb{Z}_2)$ of the stable homeomorphism space

Question

Let $TOP$ be the stable homeomorphism space, with $TOP(n) = Homeomorphisms(\mathbb{R}^n)$. What is known about its $\mathbb{Z}_2$-homology $H_*(TOP, \mathbb{Z}_2)$? In particular I am interested in the map $H_*(TOP, \mathbb{Z}_2) \rightarrow H_*(G, \mathbb{Z}_2)$, where $G$ is the stable space of homotopy automorphisms of spheres.

Can $H_*(TOP, \mathbb{Z}_2)$ be computed knowing $H_*(G, \mathbb{Z}_2)$, $H_*(G/TOP, \mathbb{Z}_2)$ or $H_*(BTOP, \mathbb{Z}_2)$?

My specific question: Is the map $H_{4k+2}(TOP, \mathbb{Z}_2) \rightarrow H_{4k+2}(G, \mathbb{Z}_2)$ injective on the image of $\pi_{4k+2}(TOP)$ under the Hurewicz homomorphism? So for example, is it injective on primitives? Does somebody have an idea or a reference I could look at?

Even more specific (and also sufficient): Let $\kappa \in \pi_{14}^s = \pi_{14}G$ be the element with Kervaire invariant 0. It lifts to $\pi_{14}(TOP)$. Is its image under $h : \pi_{14}(TOP) \rightarrow H_{14}(TOP, \mathbb Z_2)$ zero? (By Lemma 4.3 of arxiv.org/abs/1504.06752v2, $h(\kappa)=0$ in $H_{14}G$.)

Answer 1

I first comment on the specific case on the kernel of $H_{14}TOP\to H_{14}G$ when restricted to the image of Hurewicz homomorphism $\pi_{14}TOP\to H_{14}TOP$ (all homology groups are $\mathbb{Z}/2$-homology). I think it is a consequence of Sullivan's decomposition that at the prime $2$ the space $G/TOP$ decomposes as a product of Eilenberg-Moore spaces $K(\mathbb{Z}/2,4n-2)$ and $K(\mathbb{Z},4n)$ (for instance see Madsen's paper projecteuclid.org/download/pdf_1/euclid.pjm/1102868638).

Serre's exact sequence of homotopy groups for the fibration $$\Omega(G/TOP)\to TOP\to G\to G/TOP\to BTOP\to BG$$ yields $$\pi_{14}\Omega(G/TOP)\to\pi_{14}TOP\to\pi_{14}G\to\pi_{14}G/TOP$$ which reads as $$0\to\pi_{14}TOP\to\mathbb{Z}/2\{\sigma^2,\kappa\}\to\mathbb{Z}/2.$$ For the Hurewicz map $h:\pi_{14}G\to H_{14}G$ it is known that $h(\sigma^2)\neq 0$ whereas $h(\kappa)=0$.

It is known that there is a Kervaire invariant one element if its Hurewicz image maps nontrivially under $H_{2^i-2}G\to H_{2^i-2}G/TOP$. As $\sigma^2$ is a Kervaire invariant one element, we then deduce that $\sigma^2$ maps nontrivially under $\pi_{14}G\to\pi_{14}G/TOP$. Therefore, it is $\kappa$ which is detected by the map $TOP\to G$. So, $h(\kappa)=0$ only tells you that $$\mathbb{Z}/2\{\kappa\}\simeq\pi_{14}TOP\to H_{14}TOP\to H_{14}G$$ is trivial. Now, I suspect that $\kappa$ maps trivially under $$\pi_{14}TOP\to H_{14}TOP$$ which is what you asked for.

EDIT(added on 28th of March) I suggest the following route to prove that $\kappa$ maps trivially under $\pi_{14}TOP\to H_{14}TOP$. Note that here $\kappa\in\pi_{14}TOP$ is any element which maps to $\kappa\in\pi_{14}G$ where we have some abuse of notation here. Also, note that $\kappa$ really lives in $\pi_{14}SG$ where $SG=Q_1S^0$ which is homotopy equivalent to $Q_0S^0$. Furthermore, note that the Hopf invariant one element $\nu\in\pi_3^s\simeq\pi_3G$ pulls back to $\pi_3TOP$ and I think we can show it is a unique pull back (this uniqueness helps in showing that a triple Toda bracket in $\pi_*TOP$ is defined). Now, following arguments of Lemma 4.3 of https://arxiv.org/pdf/1504.06752v2.pdf we may construct $\kappa$ as an unstable map as a triple Toda bracket associated to $$S^{13}\stackrel{\beta}{\longrightarrow}\Gamma^6(\Sigma^4K)\stackrel{\Gamma^6\alpha}{\longrightarrow}\Gamma^6 S^3\stackrel{\nu}{\longrightarrow}Q_0S^0$$ where $\Gamma^6=\Omega^6\Sigma^6$. The epimorphism $\pi_3TOP\to\pi_3G\simeq\pi_3SG$ then allows to construct a triple Toda bracket for an element in $\pi_{14}TOP$ which maps to $\kappa$ with trivial indeterminacy, hence representing a generator of $\pi_{14}TOP$. A composition of the form $$S^{14}\stackrel{\beta^{\flat}}{\longrightarrow}C_{\Gamma^6\alpha}\stackrel{\nu_\sharp}{\longrightarrow}TOP$$ represents the element constructed as this triple Toda bracket. Now, by the same arguments in Lemma 4.3 https://arxiv.org/pdf/1504.06752v2.pdf we have $(\beta^\flat)_*=0$ which shows that the element $\kappa\in\pi_{14}TOP$ acts trivially in homology which is the desired result.

Answer 2

You should certainly start by looking at the book "The classifying spaces for surgery and cobordism of manifolds" by Madsen and Milgram. There is a PDF copy at http://www.maths.ed.ac.uk/~aar/papers/madmil.pdf. That contains more discussion of $BTop$ and $G/Top$ than $Top$ itself, but perhaps that will be enough for you. The books "$E_\infty$ ring spaces and $E_\infty$ ring spectra" (http://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf; May, Quinn, Ray and Tornehave) and "Homology of iterated loop spaces" (http://www.math.uchicago.edu/~may/BOOKS/homo_iter.pdf, around page 162; Cohen, Lada and May) may also be useful. If you have looked at some of these already, then you should add details to the question, clarifying how they fall short of what you need.