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