Chern-Gauss-Bonnet and Lefschetz Duality from a currential point of view

TL;DR

This paper reveals that the Chern-Gauss-Bonnet theorem for manifolds with boundary can be understood as Lefschetz Duality, using the language of currents and relative cohomology, and introduces new forms related to the zero section.

Contribution

It provides a currential perspective on classical theorems, connecting the Chern-Gauss-Bonnet theorem with Lefschetz Duality and introducing secondary transgression forms for odd rank bundles.

Findings

Shows the Chern-Gauss-Bonnet theorem as Lefschetz Duality via currents.

Introduces secondary transgression forms for odd rank bundles.

Connects Thom isomorphism with relative de Rham cohomology.

Abstract

We use the mapping cone for the relative deRham cohomology of a manifold with boundary in order to show that the Chern-Gauss-Bonnet Theorem for oriented Riemannian vector bundles over such manifolds is a manifestation of Lefschetz Duality in any of the two embodiments of the latter. We explain how Thom isomorphism fits into this picture, complementing thus the classical results about Thom forms with compact support. When the rank is odd, we construct, by using secondary transgression forms introduced here, a new closed pair of forms on the disk bundle associated to a vector bundle, pair which is Lefschetz dual to the zero section.