Differential K-theory and localization formula for $\hat{I}\cdot$-invariants

Abstract

In this paper, we obtain a localization formula in differential K-theory for $S^1$-action. Then by combining an extension of Goette's result on the comparison of two types of equivariant $\hat{I}\cdot$-invariants, we establish a version of localization formula for equivariant $\hat{I}\cdot$-invariants. An important step of our approach is to construct a pre-$\hat{I}»$-ring structure in differential K-theory which is interesting in its own right.

51 pages. Final version, to appear in Invent. Math