The universal eta-invariant for manifolds with boundary

TL;DR

This paper extends the universal eta-invariant to manifolds with boundaries, enabling new secondary invariants and providing a new interpretation of Laures' f-invariant, while also improving related modularity results.

Contribution

It introduces a generalized universal eta-invariant for manifolds with boundary and interprets Laures' f-invariant within this framework.

Findings

Construction of secondary descendants of the eta-invariant

Interpretation of Laures' f-invariant as a special case

Improved modularity result for eta invariants

Abstract

We extend the theory of the universal eta-invariant to the case of relative bordism groups of manifolds with boundaries. This allows the construction of secondary descendants of the universal eta-invariant. We obtain an interpretation of Laures' f-invariant as an example of this general construction. As an aside we improve a recent result of Han-Zhang on the modularity of a certain power series of eta invariants.