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

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.

64 pages, 3 figures, this paper is merged into Signature and Toledo invariants for flat unitary bundles over surfaces with boundary; arXiv:2202.06436

References in corpus (3)

• Density of Zariski density for surface groups
• Index type invariants for twisted signature complexes and homotopy invariance
• Two papers which changed my life: Milnor's seminal work on flat manifolds and flat bundles