# Source request for $H^*(B\mathrm{TOP},\mathbb{Q}) \cong H^*(BO,\mathbb{Q})$

Let $B\mathrm{TOP}$ denote the classifying space for microbundles, i.e. $B\operatorname{Homeo}(\mathbb{R}^n,0)$. Now we get a map from $BO$ to $B\mathrm{TOP}$ via the inclusion. Let $f$ denote the corresponding map in the rational cohomology rings.

Andrew Ranicki claims in his paper "On the construction and topological invariance of the Pontryagin classes" that the surjectivity of $f$ is equivalent to the topological invariance of the Pontryagin numbers, which is fine. But then he also writes without a source, that it is now known, that these two rings are actually isomorphic.

So on the one hand I would like to get a source for this "fact" on the other hand I would like to know if there is some nice meaning behind the injectivity of this map, like for surjectivity. I mean there is the obvious translation, that whenever to rational characteristic classes for micro bundles agree on vector bundles then they have to be the same, but maybe there is a stronger statement.

Link to the paper: http://www.maths.ed.ac.uk/~aar/papers/invtop.pdf p.311

1. The canonical map $BPL\rightarrow BTOP$ is a rational homotopy equivalence by works of Thom, Novikov, Kirby-Siebenmann and others. In fact the homotopy fiber $TOP/PL$ is a $K(\mathbb{Z}/2,3)$. A nice reference is the survey "Piecewise Linear Structures on Topological Manifolds" by Rudyak.
2. The canonical map $BO\rightarrow BPL$ is also a rational homotopy equivalence, this follows from smoothing theory. And the $i$-th homotopy groups of $PL/O$ are equal to zero when $i\leq 5$ isomorphic to $\Theta_i$ when $i\geq 6$. Where $\Theta_i$ is the group of exotic spheres: the equivalence classes of smoothing on $S^i$ under orientation-preserving diffeomorphism. As $\Theta_i$ are finite abelian groups you get your answer.