Signature, Toledo and eta invariants for surface group representations in the real symplectic group
Inkang Kim1 ; Pierre Pansu2 ; Xueyuan Wan3
1 School of Mathematics, KIAS, Heogiro 85, Dongdaemun-gu Seoul, 02455 (Republic of Korea)
2 Universit Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay 91405 Orsay Cédex (France)
3 Mathematical Science Research Center Chongqing University of Technology, Chongqing 400054 (China)
Séminaire de théorie spectrale et géométrie, Tome 37 (2021-2022), pp. 1-17.

In this paper, by using the 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 ρ 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 surfaces with boundary.

Publié le :
DOI : 10.5802/tsg.381
Classification : 14J60, 58J20, 58J28
Keywords: Signature, Toledo invariant, surface group representation, real symplectic group, eta invariant, Rho invariant

Inkang Kim 1 ; Pierre Pansu 2 ; Xueyuan Wan 3

1 School of Mathematics, KIAS, Heogiro 85, Dongdaemun-gu Seoul, 02455 (Republic of Korea)
2 Universit Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay 91405 Orsay Cédex (France)
3 Mathematical Science Research Center Chongqing University of Technology, Chongqing 400054 (China) 
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     title = {Signature, {Toledo} and eta invariants for surface group representations in the real symplectic group},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
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Inkang Kim; Pierre Pansu; Xueyuan Wan. Signature, Toledo and eta invariants for surface group representations in the real symplectic group. Séminaire de théorie spectrale et géométrie, Tome 37 (2021-2022), pp. 1-17. doi : 10.5802/tsg.381. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.381/

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