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](https://arxiv.org/abs/1504.06752v2), $h(\kappa)=0$ in $H\_{14}G$.)



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


