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

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. s paper, we focus on the fundamental group, π 1 . When this group is finite or polycyclic-by-finite, we prove that if the π 1 -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 π 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 π 1 ( M k f ( X ) ) is either trivial or isomorphic to Z 2 .