Infinite-Dimensional Degree Theory and Ramer’S Finite Co-Dimensional Differential Forms

Abstract

Infinite-dimensional degree theory, especially for Fredholm maps with positive index as developed with Tromba, is combined with Ramer’s unpublished thesis work on finite co-dimensional differential forms. As an illustrative example, the approach of Nicolaescu and Savale to the Gauss–Bonnet–Chern theorem for vector bundles is reworked in this framework. Other examples mentioned are Kokarev and Kuksin’s approach to periodic differential equations and to forced harmonic maps. A discussion about how such forms and their constructions and cohomology relate to constructions for diffusion measures on path and loop spaces is also included.