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

Inkang Kim; Pierre Pansu; Xueyuan Wan

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arXiv · 2109.08952

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

Inkang Kim·Pierre Pansu·Xueyuan Wan

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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.

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BibTeX RIS

Inkang Kim, Pierre Pansu, Xueyuan Wan. 2021-09-18. Signature, Toledo invariant and surface group representations in the real symplectic group. https://arxiv.org/abs/2109.08952

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