Abstract
Suppose W is a 4-manifold with good fundamental group and M is a closed simply-connected 4-manifold. Suppose we are given two decompositions h 1: W ⋍ M# W 1 and h 2: W ⋍ M# W 2 inducing the same decomposition of π 2 W. In this paper we study when we can conclude that W 1 and W 2 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.
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