Derham’s theorem for simplicial complexes

Abstract

This chapter begins with a definition of the piecewise linear rational polynomial forms on a simplicial complex. These are shown to compute the usual rational singular cohomology (the p.l. deRham theorem). The construction is shown to be natural under subdivision of the simplicial complex and to produce a multiplicative isomorphism to singular cohomology. Connections with the usual smooth deRham theorem are given using piecewise smooth forms on a smoothly triangulated manifold.