Is there a Whitney sum formula for topological rational Pontryagin classes? I thought the answer is yes, but now I cannot find a reference. Is it even true? The PL case would also be of interest.



Whitney sum formula for topological Pontryagin classes
Yes. A simple argument is that $BO \to BTOP$ is a rational equivalence and an H-space map (in fact even an infinite loop map), so it follows from the Whitney sum formula for vector bundles.


Edit: The argument for this is as follows. Let $\mu : BTOP \times BTOP \to BTOP$ be the map corresponding to Whitney sum of (stable, topological) bundles. The question is whether the identity
$$\mu^\* p\_n = \sum\_{ a + b = n} p\_a \otimes p\_b$$
holds. As the map $BO \to BTOP$ is a rational equivalence and an H-space map, it is equivalent to verify this equation in the cohomology of $BO$ instead, where it indeed holds.


