Vol. 296, No. 1, 2018

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Higgs bundles over cell complexes and representations of finitely presented groups

Georgios Daskalopoulos, Chikako Mese and Graeme Wilkin

Vol. 296 (2018), No. 1, 31–55

The purpose of this paper is to extend the Donaldson–Corlette theorem to the case of vector bundles over cell complexes. We define the notions of a vector bundle and a Higgs bundle over a complex, and describe the associated Betti, de Rham and Higgs moduli spaces. The main theorem is that the SL(r, ) character variety of a finitely presented group Γ is homeomorphic to the moduli space of rank-r Higgs bundles over an admissible complex X with π1(X) = Γ. A key role is played by the theory of harmonic maps defined on singular domains.

Higgs bundles, harmonic maps, simplicial complexes
Mathematical Subject Classification 2010
Primary: 58E20
Secondary: 53C07, 58D27
Received: 3 December 2016
Revised: 8 February 2018
Accepted: 23 February 2018
Published: 1 May 2018
Georgios Daskalopoulos
Department of Mathematics
Brown University
Providence, RI
United States
Chikako Mese
Department of Mathematics
Johns Hopkins University
Baltimore, MD
United States
Graeme Wilkin
Department of Mathematics
National University of Singapore