From: Linus Kramer Subject: Re: Pöncaré_Hopf Date: Tue, 01 Feb 2000 09:56:06 +0100 Newsgroups: sci.math.research Summary: [missing] philippe bich wrote: > > Does anyone know a generalisation of Poincaré-hopf ? > More precisely, if M is a non-differentiable compact manifold of non-zero > euler characteristic, > and if we have a bundle on M (not only a vector bundle) , does there exist > minimal conditions that assure that every section of the bundle > has null-point ? (excuse my english...) I assume that you refer to the following result: if M is a closed orientable smooth manifold of positive Euler characteristic, then every smooth tangent vector field has a 0. This is certainly true for topological manifolds. The substitute for the tangent bundle is Milnor's tangent microbundle: the canonical map \pi: M x M --> diag(M), (p,q) --> (p,p). By Kister's microbundle representation theorem, there exists a small open neighborhood U of the diagonal in M x M such that the map \pi, restricted to this neighborhood, is an honest fibre bundle with fibre R^n (n=the dimension of the manifold). In case that M is smooth, U is just a tubular neighborhood of the diagonal. Ok, so I assume the question is if the map \pi|U admits a right inverse s, \pi s(p,p)=(p,p), such that s(p,p) is different from (p,p) for all p in M. If such a map exists, then the Gysin sequence (see Spanier's book) shows that the map x maps to x \cup e is trivial on the cohomology of M (e is the Euler class of M - it lives in dimension n and is a multiple of the fundamental cohomology class.) But then 1 \cup e = 0, hence e = 0. So, a topological manifold of positive Euler characteristic cannot have a continuous nonzero tangent vector field. Regards, Linus Kramer PS: More results about continuous vector fields on topological manifolds can be found in Stern, Ronald J. On topological and piecewise linear vector fields. Topology 14 (1975), no. 3, 257--269. -- Linus Kramer Mathematisches Institut Universitaet Wuerzburg Am Hubland 97074 Wuerzburg Germany E-mail: kramer@mathematik.uni-wuerzburg.de http://www.mathematik.uni-wuerzburg.de/~kramer