Direct image for multiplicative and relative K-theories from transgression of the families index theorem, part 3

TL;DR

This paper completes a series on the direct image in multiplicative and relative K-theories, establishing functoriality for double submersions within these frameworks.

Contribution

It extends the theory of direct images to relative and nonfree multiplicative K-theory, proving functoriality for double submersions.

Findings

Proves functoriality of direct image for relative K-theory.

Addresses double fibration in multiplicative K-theories.

Completes the theoretical framework initiated in earlier parts.

Abstract

This is the final part of the work started in math.DG/0611281 and math.DG/0703916. Here the question of double fibration ois adressed both for relative k-theory and free multiplicative K-theory. In the case of relative and ``nonfree'' multiplicative K-theory, the direct image is proved to be functorial for double submersions.