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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

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  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.

3.The situation is completely different in the unstable range. A nice survey is "Dalian notes on rational Pontryagin classes" by Michael Weiss.

Refs:

  • The Hauptvermutung book (Kluwer 1996).

  • M. Hirsch and B. Mazur "Smoothings of piecewise linear manifolds", Annals of Mathematics Studies 80. (Princeton University Press, Princeton, 1974).

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