A response to the question about Euler classes, which appears at the
bottom of the page.....DMD
__________________________________________

Date: Fri, 05 Jan 2001 11:40:52 +0000
From: Andrew Baker <a.baker@maths.gla.ac.uk>
Subject: Fwd: question abt Euler class

Isn't this just Hopf invariant 1?  Clearly only dim bundle = n is interesting.

If you take the Thom complex Th you get a 2-cell complex with cohomology
generators
u_n, u_{2n} and then in the usual H^*(S^n)-module structure of H^*(Th) the
relation
u_n^2=eu_n holds. Also, multiplication by u_n gives an iso
H^k(S^n)->H^{k+n}(Th).

In mod 2 cohomology this gives
Sq^nu_n=eu_n.
Hopf invariant 1 now says that for n even, you can only get a non-zero lhs
when n=2,4,8

As to a reference, try Milnor & Stasheff or Steenrod & Epstein.

Andy Baker


>Envelope-to: a.baker@maths.gla.ac.uk
>Date: Thu, 04 Jan 2001 13:31:21 EST
>From: dmd1@Lehigh.EDU (DON DAVIS)
>X-Mailer: SENDM [Version 2.0.17]
>Subject: question abt Euler class
>To: Distribution.List@lehigh.edu (toplist)
>
>
>Date: Jan 4, 2001
>From: Gerard Walschap <gerard@math.ou.edu>
>Subject: ref about Euler classes
>
>         I apologize for sending the same message I did earlier. The
>thing is our server went down for about 10 days, and any reply would
>have bounced back.
>
>         Is there any reference for the following fact (for which I
>have a simple proof): If n is not 2, 4, or 8, then the Euler class of
>any bundle over the n-sphere is an even multiple of a generator of
>H^n(S^n).
>
>
>         Gerard Walschap <gerard@math.ou.edu>