Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Consider a simply connected, smooth, projective, complex surface X. Let M-k(f) (X) be the moduli space of framed irreducible anti-self-dual connections on a principal SU(2)-bundle over X with second Chern class k > 0, and let C-k(f) (X) be the corresponding space of all framed connections, modulo gauge equivalence. A famous conjecture by M. Atiyah and J. Jones says that the inclusion map M-k(f) (X) -> C-k(f) (X) induces isomorphisms in homology and homotopy through a range that grows with k. In this paper, we focus on the fundamental group, pi(1). When this group is finite or polycyclic-by-finite, we prove that if the pi(1)-part of the conjecture holds for a surface X, Keywords: then it also holds for the surface obtained by blowing up X at n points. As a corollary. Anti-self-dual connections we get that the pi(1)-part of the conjecture is true for any surface obtained by blowing up n times the complex projective plane at arbitrary points. Moreover, for such a surface, the fundamental group pi(1)(M-k(f)(X)) is either trivial or isomorphic to Z(2). (C) 2011 Elsevier B.V. All rights reserved.1593633645MIURGNSAGA of INdAM (Italy)International Centre for Mathematical Sciences (ICMS) in EdinburghInternational Centre for Mathematics Research (CIRM), Foundation Bruno Kessler, in TrentoFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)