On the geometry of moduli spaces of anti-self-dual connections

Consider a simply connected, smooth, projective, complex surface X. Let 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 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 induces isomorphisms in homology and homotopy through a range that grows with k.In this paper, we focus on the fundamental group, π1. When this group is finite or polycyclic-by-finite, we prove that if the π1-part of the conjecture holds for a surface X, then it also holds for the surface obtained by blowing up X at n points. As a corollary, we get that the π1-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 is either trivial or isomorphic to Z2.