Homology of the completion of instanton moduli spaces

Abstract

Let $M (k,G)$ be the moduli space of based gauge equivalence classes of $G$-instantons on $S^4$ with instanton number $k$. $M(k, G)$ has the Uhlenbeck completion $\overline{M} (k,G) = \bigcup_{q=0}^k \mathrm{SP}^q (\mathbb R^4) \times M (k-q,G)$, where $\mathrm{SP}^q (\mathbb R^4)$ denotes the $q$-fold symmetric product of $\mathbb R^4$. Let $X (k,G)$ be the first two strata of the completion: $X (k,G) = M (k,G) \cup \mathbb R^4 \times M (k-1,G)$. In this paper we study the homology of $X (k,G)$ for $G= SU(n)$ or $Sp(n)$, and relate this to the homology of a certain homotopy theoretic fibre.