Relative Index Pairing and Odd Index Theorem for Even Dimensional Manifolds

TL;DR

This paper extends index theory to even-dimensional manifolds, establishing a new eta invariant relation and an odd-dimensional counterpart for boundary manifolds, enriching the understanding of index pairings.

Contribution

It introduces an analogue of the Atiyah-Patodi-Singer index theorem for even-dimensional manifolds and relates eta invariants, also providing an odd-dimensional boundary counterpart.

Findings

Eta invariant matches Dai and Zhang's invariant up to an integer.

Established an even-dimensional index pairing analogue.

Derived an odd-dimensional boundary index pairing counterpart.

Abstract

We prove an analogue for even dimensional manifolds of the Atiyah-Patodi-Singer twisted index theorem for trivialized flat bundles. We show that the eta invariant appearing in this result coincides with the eta invariant by Dai and Zhang up to an integer. We also obtain the odd dimensional counterpart for manifolds with boundary of the relative index pairing by Lesch, Moscovici and Pflaum.