On the fundamental groups of moduli spaces of irreducible SU(2)-connections over closed 4-manifolds (Q917094)

Let M be a connected oriented smooth closed 4-manifold and \(P_ k\to M\) be a principal SU(2) bundle over M with \(c_ 2=k\). We denote by \({\mathcal A}_ k\) the space of all smooth connections on \(P_ k\) and by \({\mathcal A}^*_ k\) the subspace of \({\mathcal A}_ k\) whose elements are irreducible connections. Let \({\mathcal G}_ k\) be a (full) gauge group of \(P_ k\) that is \(Aut(P_ k)\). We set \({\mathcal B}_ k={\mathcal A}_ k/{\mathcal G}_ k\) and \({\mathcal B}^*_ k={\mathcal A}_ k^*/{\mathcal G}_ k\). We call \({\mathcal B}^*_ k\) the moduli space of irreducible SU(2)-connections on \(P_ k\). We calculate the fundamental group of \({\mathcal B}^*_ k\). Then we obtain the following: Theorem. Let M be a closed 4-manifold as above. Suppose that M is simply connected. (1) When the intersection form of M is of odd type, then \[ \pi_ 1({\mathcal B}^*_ k)=1. \] (2) When the intersection form of M is of even type, then \[ \pi_ 1({\mathcal B}^*_ k)=1\text{ if }c_ 2(P_ k)=k\text{ is odd.} \] \[ \pi_ 1({\mathcal B}^*_ k)={\mathbb{Z}}/2{\mathbb{Z}}\text{ if } c_ 2(P_ k)=k\text{ is even.} \] By \textit{S. K. Donaldson} [J. Differ. Geom. 18, 279-315 (1983; Zbl 0507.57010)] we know that \(\pi_ 1({\mathcal B}^*_ 1)=1\). We note that since \({\mathcal G}_ k\) acts on \({\mathcal A}^*_ k\) not freely (indeed each irreducible connection has the isotropy subgroup \({\mathbb{Z}}/2{\mathbb{Z}})\), the topology of \({\mathcal B}^*_ k\) is not simply deduced from that of the gauge group \({\mathcal G}_ k\).