# Journal of Noncommutative Geometry

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**Volume 10, Issue 1, 2016, pp. 265–306**

**DOI: 10.4171/JNCG/234**

Published online: 2016-03-22

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

Alexander Gorokhovsky^{[1]}, Hitoshi Moriyoshi

^{[2]}and Paolo Piazza

^{[3]}(1) University of Colorado at Boulder, USA

(2) Nagoya University, Japan

(3) Università di Roma La Sapienza, Italy

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.

*Keywords: *Galois coverings, groupoids, group cocycles, index classes, relative pairing, excision, Atiyah–Patodi–Singer higher index theory, higher eta invariants

Gorokhovsky Alexander, Moriyoshi Hitoshi, Piazza Paolo: A note on the higher Atiyah–Patodi–Singer index theorem on Galois coverings. *J. Noncommut. Geom.* 10 (2016), 265-306. doi: 10.4171/JNCG/234