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

@inproceedings{Kim2021SignatureTI, title={Signature, Toledo invariant and surface group representations in the real symplectic group}, author={Inkang Kim and Pierre Pansu and Xueyuan Wan}, year={2021} }

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… Expand

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