Periodic Index Theory and Equivariant Torus Signature

TL;DR

This paper develops an index jump formula for elliptic complexes on end-periodic manifolds, introduces the periodic rho invariant, and connects it with spectral flow, with applications to signature invariants and knot theory.

Contribution

It generalizes index theory to end-periodic manifolds, defines the periodic rho invariant, and relates different signature invariants and invariants in knot theory.

Findings

Derived an index jump formula for elliptic complexes.

Defined the periodic rho invariant and linked it to spectral flow.

Proved equivalence of signature invariants and a surgery formula for the Furuta-Ohta invariant.

Abstract

We deduce an index jump formula for first order elliptic complexes over end-periodic manifolds, which generalizes the corresponding result for the DeRham complex. In the case of the anti-self-dual DeRham complex, we define the periodic rho invariant for a class of $4$-manifolds, and identify it with the periodic spectral flow of this complex. As an application, we prove the equivalence (under a mild homological assumption) of two signatures invariants defined by means of Yang-Mills theory and geometric topology respectively for essentially embedded tori in homology $S^1 \times S^3$. We also prove a surgery formula for the singular Furuta-Ohta invariant, which corresponds to a potential exact triangle of singular instanton homology for knots.