#### Vol. 296, No. 1, 2018

 Recent Issues Vol. 305: 1  2 Vol. 304: 1  2 Vol. 303: 1  2 Vol. 302: 1  2 Vol. 301: 1  2 Vol. 300: 1  2 Vol. 299: 1  2 Vol. 298: 1  2 Online Archive Volume: Issue:
 The Journal Editorial Board Subscriptions Officers Special Issues Submission Guidelines Submission Form Contacts ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Author Index To Appear Other MSP Journals
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
##### Abstract

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 $SL\left(r,ℂ\right)$ character variety of a finitely presented group $\Gamma$ is homeomorphic to the moduli space of rank-$r$ Higgs bundles over an admissible complex $X$ with ${\pi }_{1}\left(X\right)=\Gamma$. A key role is played by the theory of harmonic maps defined on singular domains.

##### Keywords
Higgs bundles, harmonic maps, simplicial complexes
##### Mathematical Subject Classification 2010
Primary: 58E20
Secondary: 53C07, 58D27