## Homotopy type of TOP(4)/PL(4)

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).

-
What is $TOP(n)$? – Steven Landsburg Sep 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: tinyurl.com/cddp8oq – Benjamin Dickman Sep 12 at 6:32
@Steven Landsburg: From books.google.it/… 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 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 at 15:35
$\text{TOP}(n)$ is in fact the group of self-homeomorphisms of ${\mathbb R}^n$, as Qfwfq states. – Ricardo Andrade Sep 12 at 19:04