A Geometric Definition of the Integral and Applications

Abstract

The standard definition of integration of differential forms is based on local coordinates and partitions of unity. This definition is mostly a formality and not used used in explicit computations or approximation schemes. We present a definition of the integral that uses triangulations instead. Our definition is a coordinate-free version of the standard definition of the Riemann integral on $\mathbb{R}^n$ and we argue that it is the natural definition in the contexts of Lie algebroids, stochastic integration and quantum field theory, where path integrals are defined using lattices. In particular, our definition naturally incorporates the different stochastic integrals, which involve integration over H\"{o}lder continuous paths. Furthermore, our definition is well-adapted to establishing integral identities from their combinatorial counterparts. Our construction is based on the observation that, in great generality, the things that are integrated are determined by cochains on the pair groupoid. Abstractly, our definition uses the van Est map to lift a differential form to the pair groupoid. Our construction suggests a generalization of the fundamental theorem of calculus which we prove: the singular cohomology and de Rham cohomology cap products of a cocycle with the fundamental class are equal.