Subject: Re: one posting From: Donald Kahn Date: Fri, 30 Dec 2005 23:26:27 -0600 the work of wu wen-tsun, on this subject, is published in scientia sinica in the 1950's, and is available as a translation from the amer. math. society. don kahn On Dec 30, 2005, at 9:36 AM, Don Davis wrote: > Only one posting this week.........DMD > ____________________________________________________________ > > Subject: Re: question about Pontryagin classes > From: Yuli Rudyak > Date: Thu, 22 Dec 2005 19:51:19 -0500 > > Response to: > > > > Subject: char classes > > From: wolfgang Ziller > > Date: Tue, 20 Dec 2005 23:39:26 -0200 > > > > I would like to know what kind of information is > > contained in the Pontryagin classes that are homotopy invariants. > > I know a theorem by Hirzebruch that p_1(M) mod 24 > > is a homotopy invariant. > > What other combinations like this, mod some integer, > > are homotopy invariants. > > Any that involve only p_1 and p_2 ? > > Any references would be welcome. > > > > wolfgang ziller > > > > First, $p_i\mod 2=w_{2i}^2$, and so the classes $p_i\mod 2$ are homotopy invariant. > > Concerning modulo $p$ with $p$ odd prime. I think, Wu has some results about it. > For example, consider the Steenrod power operation $P_^$ of degree $2i(p-1)$ and the charactersitic class $a(\xi)=\phi^{-1}P^iu_{\xi}$ where $u$ is the > Thom class mod $p$ of the oriented vector bundle $\xi$ and $\phi$ is the Thom > isomorphism. This class is clearly a homotopy invariant, and it must be a > polynomial of Pontryagin classes mod $p$, since the mod-p cohomology of $BSO$ are polynomials of Pontryagin classes. > > Yuli Rudyak > >