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 ____________________________________________________________________________ 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 __________________________________________________________________________ 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