Signature and Toledo invariants for flat unitary bundles over surfaces with boundary

Kim Inkang , Pierre Pansu (1) , Xueyuan Wan

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

This paper deals with the representations of the fundamental groups of compact surfaces with boundary into classical simple Lie groups of Hermitian type. We relate work on the signature of the associated local systems, due to Lustig, Meyer and Atiyah, to Burger-Iozzi-Wienhard's Toledo invariant. To measure the difference, we extend Atiyah-Patodi-Singer's rho invariant, initially defined on U(p), to discontinuous class functions, first on U(p,q), and then on other classical groups via embeddings into U(p,q). As an application, we obtain a Milnor-Wood type inequality which slightly differs from, and sometimes improves upon Burger-Iozzi-Wienhard's version.

Mots clés: signature; Toledo invariant; surface group representations; eta invariant; rho invariant; unitary group; classical bounded symmetric domains; Milnor-Wood inequalities; 2020 Mathematics Subject Classification. 14J60; 58J20; 58J28

Domaines: Géométrie différentielle [math.DG]

Kim Inkang, Pierre Pansu, Xueyuan Wan. Signature and Toledo invariants for flat unitary bundles over surfaces with boundary. 2022. ⟨hal-03565594⟩