Signature, Toledo invariant and surface group representations in the real symplectic group
Geometric Topology
2022-03-02 v2 Differential Geometry
Representation Theory
Spectral Theory
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.
Keywords
Cite
@article{arxiv.2109.08952,
title = {Signature, Toledo invariant and surface group representations in the real symplectic group},
author = {Inkang Kim and Pierre Pansu and Xueyuan Wan},
journal= {arXiv preprint arXiv:2109.08952},
year = {2022}
}
Comments
64 pages, 3 figures, this paper is merged into Signature and Toledo invariants for flat unitary bundles over surfaces with boundary; arXiv:2202.06436