Homotopy invariance of 4-manifold decompositions: Connected sums

Given any homotopy equivalence f : M → X 1 # ⋯ # X n of closed orientable 4-manifolds, where each fundamental group π 1 ( X i ) satisfies Freedmanʼs Null Disc Lemma, we show that M is topologically h-cobordant to a connected sum M ′ = M 1 ′ # ⋯ # M n ′ such that f is h-bordant to some f 1 ′ # ⋯ # f n ′ with each f i ′ : M i ′ → X i 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. e establish that the Borel Conjecture is true in dimension 4, up to s-cobordism, if the fundamental group satisfies the Farrell–Jones Conjecture.