Version nr. 2 includes improved results for moduli of representations in GL(n,C), provided proof of Proposition 4.13 (4.12 of v1) correcting argument given in earlier paper, other minor corrections. Version nr. 3 contains corrected Remark 4.6 and other minor corrections. [Present] Version nr. 4 includes corrected statements of Proposition 4.17 and Theorems 4.19 and 4.22 (former Theorems 4.18 and 4.20), and is to appear in Topology.We calculate certain homotopy groups of the moduli spaces for representations of a compact oriented surface in the Lie groups GL(n,C) and U(p,q). Our approach relies on the interpretation of these representations in terms of Higgs bundles and uses Bott–Morse theory on the corresponding moduli spaces.Second and Third authors partially supported by Ministerio de Educación y Ciencia and Conselho de Reitores das Universidades Portuguesas through Acción Integrada Hispano-Lusa HP-2000-0015 (Spain) / E–30/03 (Portugal). Second author partially supported by Ministerio de Educación y Ciencia (Spain) through Project BFM2000-0024. Third author partially supported by the Centro de Matemática da Universidade do Porto and the project POCTI/MAT/58549/2004, financed by FCT (Portugal) through the programmes POCTI and POSI of the QCA III (2000–2006) with European Community (FEDER) and national funds.Peer reviewed