Abstract

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.