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It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that $\mathrm{TOP}(4)/\mathrm{PL}(4)\to K({\mathbb Z}/2,3)$ is 5-connected. Is it known whether $\mathrm{TOP}(4)/\mathrm{PL}(4)$ is not equivalent to $K({\mathbb Z}/2,3)$?

Edit: The following lists the relevant definitions.

  • The topological group $\mathrm{TOP}(n)$ is the group of self-homeomorphisms of ${\mathbb R}^n$ with the compact-open topology.

  • The topological group $\mathrm{PL}(n)=|\mathrm{PL}_S(n)|$ is defined as the geometric realization of the simplicial group $\mathrm{PL}_S(n)$. The $k$-simplices of $\mathrm{PL}_S(n)$ are the piecewise linear homeomorphisms $\Delta^k\times{\mathbb R}^n\to\Delta^k\times{\mathbb R}^n$ which commute with the projection onto $\Delta^k$.

  • With the above definitions, there exists a canonical map of topological groups $\mathrm{PL}(n)\to\mathrm{TOP}(n)$. Then the space $\mathrm{TOP}(n)/\mathrm{PL}(n)$ is defined as the homotopy fibre of the induced map $B\mathrm{PL}(n)\to B\mathrm{TOP}(n)$. It is not actually a quotient of a group by a subgroup.

  • Here is a way of recovering the homotopy type of $\mathrm{TOP}(n)/\mathrm{PL}(n)$ as an actual quotient. Let $\mathrm{TOP}_S(n)$ be the singular complex of $\mathrm{TOP}(n)$: $\mathrm{TOP}_S(n)$ is the simplicial set whose $k$-simplices are continuous maps $\Delta^k\to\mathrm{TOP}(n)$; these are in canonical bijection with the homeomorphisms $\Delta^k\times{\mathbb R}^n\to\Delta^k\times{\mathbb R}^n$ commuting with the projection onto $\Delta^k$. Hence we obtain an inclusion of simplicial groups $\mathrm{PL}_S(n)\hookrightarrow\mathrm{TOP}_S(n)$, which induces by adjunction the previous map of topological groups $\mathrm{PL}(n)\to\mathrm{TOP}(n)$. The space $\mathrm{TOP}(n)/\mathrm{PL}(n)$ is weakly homotopy equivalent to the geometric realization of the simplicial set $\mathrm{TOP}_S(n)/\mathrm{PL}_S(n)$ (which is levelwise given by taking cosets).

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What is $TOP(n)$? – Steven Landsburg Sep 12 '12 at 3:20
It was unknown 20 years ago, when this question was proposed as Conjecture 3.10 in 'Differential Topology, Foliations, and Group Actions' by Paul A. Schweitzer. Link: – Benjamin Dickman Sep 12 '12 at 6:32
@Steven Landsburg: From… I gather $TOP(n)$ is the group of self-homeomorphisms of $\mathbb{R}^n$ and $PL(n)$ the subgroup of piecewise linear ones. – Qfwfq Sep 12 '12 at 9:44
@Qfwfq : This is not really my area, but I'm pretty sure that $TOP(n)$ is suppposed to classify topological microbundles on a space. This means that it is something like the classifying space for something like the group of germs of homeomorphisms of $\mathbb{R}^n$ (or maybe the pseudogroup of homeomorphisms between open sets in $\mathbb{R}^n$). Similarly for $PL(n)$. I don't have it at hand, but there are proper definitions in a book by Madsen and Milgram. – Sue Sep 12 '12 at 15:35
PL(n) needs a different definition; subspace of Top(n) won't do; see for example 1.10 in Madsen and Milgram, "The classifying spaces for surgery and cobordism of manifolds". It is BTop(n) and BPL(n) that classify the appropriately defined bundles. As far as I know the question asked is still open. In general, BTop(n) and cognate spaces are not well understood; even their cohomology is not known, although the cohomology when one goes to the limit and considers BTop and BPL is completely understood. – Peter May Feb 18 '13 at 13:59

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