Counterexamples to the Kneser conjecture in dimension four (Q1906916)

First some examples are given of closed smooth orientable 4-manifolds \(M\) whose fundamental group splits as a nontrivial free product, but such that if \(M\) is homotopy equivalent to the connected sum \(M_0 \# M_1\), then \(M_0\) or \(M_1\) is homeomorphic to \(S^4\). An example is then given of a closed smooth orientable 4-manifold \(M\) which is homeomorphic to \(M_0 \# M_1\), these being closed topological 4- manifolds, each with fundamental group cyclic of order two, but such that \(M \# k(S^2 \times S^2)\) is diffeomorphic to \(M_0 \# M_1\) for non-simply-connected smooth 4-manifolds \(M_0\) and \(M_1\) if and only if \(k \geq 8\).