Differential K-theory and localization formula for $η$-invariants

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 $η$-invariants, we establish a version of localization formula for equivariant $η$-invariants. An important step of our approach is to construct a pre-$λ$-ring structure in differential K-theory which is interesting in its own right.

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