Topology of moduli space of certain \(SU(2)\) connections of degree 2 over \(S^ 4\) (Q1199183)

In the framed moduli space of connections in the principal \(SU(2)\)-bundle over the 4-sphere with second Chern-class \(k\), the subspace \(M\) of instantons can be described by linear algebra known as the Atiyah- Drinfield-Hitchin-Manin construction. Leaving away one of their three conditions, namely the reality condition, \(M\) is naturally embedded in a certain space \(M'\) of connections. In this paper, the author computes for \(k=2\) the first two homotopy groups of \(M'\) and shows that the inclusion map from \(M\) to \(M'\) induces an isomorphism on \(Z/2\)-homology. This computes the \(Z/2\)-homology of \(M'\) since the one of \(M\) was computed by the author in a former paper. Standard methods from algebraic topology are used like the Serre spectral sequence and the Araki-Kudo operations on loop spaces.