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A Hilbert bundle description of differential K-theory

Research paper by Alexander Gorokhovsky, John Lott

Indexed on: 12 Apr '16Published on: 12 Apr '16Published in: Mathematics - Differential Geometry



Abstract

We give an infinite dimensional description of the differential K-theory of a manifold $M$. The generators are triples $[H, A, \omega]$ where $H$ is a ${\bf Z}_2$-graded Hilbert bundle on $M$, $A$ is a superconnection on $H$ and $\omega$ is a differential form on $M$. The relations involve eta forms. We show that the ensuing group is the differential K-group $\check{K}^0(M)$. In addition, we construct the pushforward of a finite dimensional cocycle under a proper submersion with a Riemannian structure. We give the analogous description of the odd differential K-group $\check{K}^1(M)$. Finally, we give a model for twisted differential K-theory.