Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Vector fields on polyhedra

Author: Michael A. Penna
Journal: Trans. Amer. Math. Soc. 232 (1977), 1-31
MSC: Primary 57D25; Secondary 58D99
MathSciNet review: 0451258
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Abstract: This paper presents a bundle theory for studying vector fields and their integral flows on polyhedra $ _ \ast $ 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.

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Keywords: Polyhedron, manifold, tangent bundle, vector field, Hopf's theorem, Gauss-Bonnet formula
Article copyright: © Copyright 1977 American Mathematical Society