From Sparks to Grundles--Differential Characters

TL;DR

This paper introduces spark complexes as a unified framework for secondary geometric invariants, demonstrating their equivalence across diverse mathematical contexts and applications in geometry, topology, and physics.

Contribution

It establishes the canonical isomorphism among various types of spark complexes and simplifies the theory of differential characters to holonomy maps.

Findings

Different spark complexes are shown to be canonically isomorphic.

Differential characters can be factored through holonomy maps.

Applications span geometry, topology, and physics.

Abstract

We introduce a new homological machine for the study of secondary geometric invariants. The objects, called spark complexes, occur in many areas of mathematics. The theory is applied here to establish the equivalence of a large family of spark complexes which appear naturally in geometry, topology and physics. These complexes are quite different. Some of them are purely analytic, some are simplicial, some are of Cech-type, and many are mixtures. However, the associated theories of secondary invariants are all shown to be canonically isomorphic. We also show that Differential characters factor to a much smaller, more geometric group, the set of holonomy maps. Numerous applications and examples are explored.