The eta invariant and equivariant bordism of flat manifolds with cyclic holonomy group of odd prime order

TL;DR

This paper computes eta invariants for certain flat spin manifolds with cyclic holonomy groups of odd prime order, revealing their triviality in equivariant spin bordism groups except in one specific case.

Contribution

It provides explicit formulas for eta invariants of flat spin manifolds with cyclic holonomy and demonstrates their triviality in reduced equivariant spin bordism groups.

Findings

Reduced eta invariant is always an integer except when p=n=3.

Explicit expressions for twisted and relative eta invariants are derived.

All such manifolds are trivial in the reduced equivariant spin bordism group, except in one case.

Abstract

We study the eta invariants of compact flat spin manifolds of dimension n with holonomy group cyclic of odd prime order p. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when p=n=3. We use the expressions obtained to show that any such manifold is trivial in the appropriate reduced equivariant spin bordism group.