Homology of the completion of instanton moduli spaces

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 (M) over bar (k, G) = boolean ORq=0k SPq (R-4) x M(k - q, G), where SPq (R-4) denotes the q-fold symmetric product of R-4. Let X(k, G) be the first two strata of the completion: X(k, G) = M(k, G) boolean OR R-4 x 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.