Subject: Re: Hopf postings and question
From: Walter Neumann
Date:Wed, 15 Aug 2007 10:26:42 -0400 (EDT)
In answer to Mark Hovey, the n-th Stiefel Whitney class is non-trival, so
there is no nonzero section.
> Subject: Question for the list
> From: Mark Hovey
> Date: Tue, 14 Aug 2007 17:06:12 -0400
>
> I have a question for the list. Let E denote the Mobius bundle over
> the circle S^1; this is the nontrivial real line bundle. Obviously E
> has no section that is everywhere nonzero, or it would be trivial. OK,
now form the product E^n = E x E x ... x E over the n-fold torus
> (S^1)^n. My (probably simple) question is this: Does E^n have a section
that is
> everywhere nonzero, or not? Thanks,
> Mark Hovey
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Subject: Re: Hopf postings and question
From: Johannes Ebert
Date: Wed, 15 Aug 2007 16:50:53 +0100 (BST)
The answer to the question about the Moebius bundle is easy. The first
Stiefel-Whitney class of E is the nontrivial element in H^1 (S^1) (with
Z/2 coefficients). Let p_i: (S^1)^n \to S^1 be the projection onto the i
th factor. Obviously E^n = p_1^* E \oplus ... p_n^* E. The Whitney sum
formula and the Kuenneth formula imply that w_n (E^n) is nonzero. Thus
there is no global section.
Best regards,
Johannes Ebert
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Subject: question
From: "Dr A. J. Baker"
Date: Wed, 15 Aug 2007 16:54:38 +0100 (BST)
Isn't the Stiefel Whitney class w_n just the product of the w_1's of
the factors (these are non-zero), hence w_n is non-zero and this is the
primary obstruction to a section. Or am I being stupid?
Andy