What is the classifying space G/Top?

Question

I simply can't find the definition(except in one book on surgery where a definition was not actually given but instead they alluded to what the definition is) and I have spent an hour and half looking. My professor used the notation when talking me today. I would be grateful for a reference. My background is evident from my stackexchange posts.

Answer 1

A reference available online is Rudyak's survey Piecewise Linear Structures on Topological Manifolds. Beware that what you are calling $G$ he calls $F$ (I think both notations are common --and awful).

$G/TOP$ can be defined as the homotopy fiber of the canonical map $BTOP \to BG$, where $BTOP$ is the classifying space for stable topological bundles and $BG$ is the classifying space for stable spherical fibrations.

$BTOP$ can be defined as follows: let $TOP_n$ denote the topological group of self homeomorphisms of $\mathbb{R}^n$ fixing the origin, and let $BTOP_n$ be its classifying space. Each $TOP_n$ can be included in the next by sending $h : \mathbb{R}^n \to \mathbb{R}^n$ to $h \times \mathrm{id} : \mathbb{R}^{n+1} \to \mathbb{R}^{n+1}$. You get induced maps $BTOP_n \to BTOP_{n+1}$ and define $BTOP$ to be the colimit of all the $BTOP_n$.

$BG$ can be defined as follows: let $G_n$ be the topological monoid of pointed homotopy self equivalences of $S^n$ and $BG_n$ its classifying space (note that homotopy equivalences don't strictly speaking have an inverse udner composition, so $G_n$ is not a group, just a monoid). Suspension gives a map $G_n \to G_{n+1}$ and you set $BG$ to the colimit of the corresponding sequence of maps $BG_n \to BG_{n+1}$.

The map $BTOP \to BG$ is induced from the operation of one-point compactification: if $h : \mathbb{R}^n \to \mathbb{R}^n$, the one-point compactification $h^\bullet : S^n \to S^n$ is a homotopy self equivalence of $S^n$.