Smooth K-Theory

TL;DR

This paper develops a new analytic model of smooth K-theory, introduces smooth K-orientations and push-forwards, and establishes a lift of the Atiyah-Singer index theorem to smooth cohomology, advancing the understanding of geometric and topological invariants.

Contribution

It constructs a multiplicative analytic model of smooth K-theory, defines smooth K-orientations and push-forwards, and lifts the Atiyah-Singer index theorem to smooth cohomology.

Findings

Constructed an analytic multiplicative model of smooth K-theory.

Defined smooth K-orientations and associated push-forward maps.

Verified the lift of the Atiyah-Singer index theorem to smooth cohomology.

Abstract

We construct an analytic multiplicative model of smooth K-theory. We further introduce the notion of a smooth K-orientation of a proper submersion and define the associated push-forward which satisfies functoriality, compatibility with pull-back diagrams, and projection and bordism formulas. We construct a multiplicative lift of the Chern character from smooth K-theory to smooth rational cohomology and verify that the cohomological version of the Atiyah-Singer index theorem for families lifts to smooth cohomology.