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

2021/09/18 by Инканг Ким, Pierre Pansu, Kim, Inkang +3
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Representation Theory (math.RT) #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2109.08952

openalex publication_date 2021/09/18 · openalex created_date 2021/09/27 · openalex updated_date 2026/07/28

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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