Homotopy invariance of 4-manifold decompositions: Connected sums

Given any homotopy equivalence f:M→X1#⋯#Xn of closed orientable 4-manifolds, where each fundamental group π1(Xi) satisfies Freedmanʼs Null Disc Lemma, we show that M is topologically h-cobordant to a connected sum M′=M1′#⋯#Mn′ such that f is h-bordant to some f1′#⋯#fn′ with each fi′:Mi′→Xi a homotopy equivalence. Moreover, such a replacement M′ of M is unique up to a connected sum of h-cobordisms. In summary, the existence and uniqueness, up to h-cobordism, of connected sum decompositions of such orientable 4-manifolds M is an invariant of homotopy equivalence.Also we establish that the Borel Conjecture is true in dimension 4, up to s-cobordism, if the fundamental group satisfies the Farrell-Jones Conjecture.