The Thom isomorphism in bivariant K-theory

TL;DR

This paper provides a simplified proof of the Thom isomorphism in bivariant K-theory for complex bundles, extending to Kasparov's KK-theory, and introduces a direct proof of Bott periodicity on R^n.

Contribution

It offers a straightforward proof of the Thom isomorphism in bivariant K-theory and Kasparov's KK-theory, including a direct proof of Bott periodicity on R^n.

Findings

Simplified proof of the Thom isomorphism for complex bundles

Extension of the Thom isomorphism to Kasparov's KK-theory

Direct proof of Bott periodicity on R^n

Abstract

We give a simple proof of the smooth Thom isomorphism for complex bundles for the bivariant K-theories on locally convex algebras considered by Cuntz. We also prove the Thom isomorphism in Kasparov's KK-theory in a form stated without proof in the conspectus. Along the way, we prove Bott periodicity directly on R^n, using for the Kasparov product the operator that also appears in recent work of Wulkenhaar on non-compact spectral triples with finite volume, and which may be seen as a unitalisation of the Dirac-element.