Talk:Pontryagin class

Stable homotopy

The article says that the Pontryagin classes generate the stable homotopy of $BO(k)$, I really don't see how that can be possible as they are cohomology classes and not homotopy classes. I understand how the Pontryagin classes are a stable cohomology class in that they pullback from $BO(k+1)$, so I think this is probably what was meant. --137.205.233.136 (talk) 12:15, 30 April 2009 (UTC)

I agree that this was intended, however it's only true modulo 2-torsion. See Milnor+Stasheff page 179.

Normalization

The old Chern-Weil formula was incorrect. For instance, it lacked the necessary factors of $\pi$. I computed the current formula myself. I'm fairly confident that it's correct, but someone else ought to verify it.

Definition

The entry starts by saying "Given a real vector bundle E over M...", but it doesn't say what M is supposed to be. — Preceding unsigned comment added by 62.189.157.194 (talk) 11:05, 14 June 2019 (UTC)

Topological Invariants

It is not correct, that Pontryagin numbers are topological invariants. For example, an orientation reversing self-diffeomorphism changes the sign of any Pontryagin number. Also it is not clear to me, why Pontryagin numbers should be integrally invariant under orientation preserving homeomorphisms, when the Pontryagin classes are not. Rationally this should be correct. — Preceding unsigned comment added by 2A00:1398:9:FB03:B46A:C6B4:FBF8:3E6E (talk) 11:12, 23 June 2020 (UTC)