A note on the higher Atiyah-Patodi-Singer index theorem on Galois coverings

TL;DR

This paper provides a new proof of the higher Atiyah-Patodi-Singer index theorem for Galois coverings with a focus on explicit formulas for higher indices associated with polynomially growing group cocycles, using relative K-theory and cyclic cohomology.

Contribution

It introduces a novel proof technique for the higher index theorem employing relative K-theory and cyclic cohomology, and explicitly formulates the higher index for polynomial growth cocycles.

Findings

Explicit formula for higher index associated with polynomial growth cocycles

New proof employing relative K-theory and cyclic cohomology

Applicable to Galois coverings with groups satisfying rapid decay condition

Abstract

Let $\Gamma$ be a finitely generated discrete group satisfying the rapid decay condition. We give a new proof of the higher Atiyah-Patodi-Singer theorem on a Galois $\Gamma$-coverings, thus providing an explicit formula for the higher index associated to a group cocycle $c\in Z^k (\Gamma;\mathbb{C})$ which is of polynomial growth with respect to a word-metric. Our new proof employs relative K-theory and relative cyclic cohomology in an essential way.