Vector fields on polyhedra

Penna, Michael A.

doi:10.1090/S0002-9947-1977-0451258-6

Vector fields on polyhedra
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by Michael A. Penna

Trans. Amer. Math. Soc. 232 (1977), 1-31

DOI: https://doi.org/10.1090/S0002-9947-1977-0451258-6

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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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