Uniqueness of connected sum decompositions in dimension 4

AbstractSuppose W is a 4-manifold with good fundamental group and M is a closed simply-connected 4-manifold. Suppose we are given two decompositions h1: W ⋍ M#W1 and h2: W ⋍ M#W2 inducing the same decomposition of π2W. In this paper we study when we can conclude that W1 and W2 are homeomorphic. As a consequence we conclude that the ∗ operation for changing the Kirby-Siebenmann invariant of a 4-manifold is well defined. We will also use this discussion to relate the ambient approach to classification to the surgery approach