Vector fields on polyhedra

Abstract

Abstract. This paper presents a bundle theory for studying vector fields and their integral flows on polyhedra^ and applications.Every polyhedron has a tangent object in the category of simplicial bundles in much the same way as every smooth manifold has a tangent object in the category of smooth vector bundles. One can show that there is a correspondence between piecewise smooth flows on a polyhedron P and sections of the tangent object of P (i.e., vector fields on P); using this result one can prove existence results for piecewise smooth flows on polyhedra. Finally an integral formula for the Euler characteristic of a closed, oriented, even-dimensional combinatorial manifold is given; as a consequence of this result one obtains a representation of Euler classes of such combinatorial manifolds in terms of piecewise smooth forms. 0. Introduction. The purpose of this paper is to describe a bundle theory for studying vector fields and their integral flows on polyhedra and, in particular, on combinatorial manifolds. Two applications of this theory are presented; these results are motivated by corresponding results for smooth manifolds.Here are the results obtained:Theorem A. The sum of the indices of any integrable vector field on a polyhedron P which has a finite number of zeroes is the Euler characteristic x(P) of P. Furthermore, if M is a closed combinatorial manifold with x(M) = 0, then there is a nowhere vanishing integrable vector field on M.Theorem A corresponds to a classical result for smooth manifolds due to Hopf (see [10]). Actually Hopfs original result (see [7]) is also valid for a particular type of vector field on a combinatorial manifold. It will be seen that even though there are similarities between Hopfs original result and Theorem A, there are also major differences. As a special case of Theorem A one also obtains an analog to a calculation of Whitney (see [14]) giving the Euler characteristic of a triangulated smooth manifold using the singularities of vector fields. Now let M be a closed, oriented, 2/i-dimensional combinatorial manifold.