Signature, Toledo invariant and surface group representations in the real symplectic group

TL;DR

This paper derives a formula linking the signature of flat symplectic bundles over surfaces to the Toledo invariant and Rho invariant, providing new inequalities and proofs for surface group representations in the real symplectic group.

Contribution

It introduces a novel formula connecting signature, Toledo invariant, and Rho invariant using index theory, and extends Milnor-Wood inequalities to surfaces with boundary.

Findings

Derived a signature formula using Atiyah-Patodi-Singer index theorem.

Established a Milnor-Wood type inequality for the signature.

Provided new proofs and modified inequalities for boundary surfaces.

Abstract

In this paper, by using Atiyah-Patodi-Singer index theorem, we obtain a formula for the signature of a flat symplectic vector bundle over a surface with boundary, which is related to the Toledo invariant of a surface group representation in the real symplectic group and the Rho invariant on the boundary. As an application, we obtain a Milnor-Wood type inequality for the signature. In particular, we give a new proof of the Milnor-Wood inequality for the Toledo invariant in the case of closed surfaces and obtain some modified inequalities for the surface with boundary.