Stable equivalence relations on 4‐manifolds

Kreck's modified surgery gives an approach to classifying smooth ‐manifolds up to stable diffeomorphism, that is, up to connected sum with copies of . In dimension 4, we use a combination of modified and classical surgery to study various stable equivalence relations which we compare to stable diffeomorphism. Most importantly, we consider homotopy equivalence up to stabilisation with copies of . As an application, we show that closed oriented homotopy equivalent 4‐manifolds with abelian fundamental group are stably diffeomorphic. We give analogues of the cancellation theorems of Hambleton–Kreck for homotopy equivalence up to stabilisations. Finally, we give a complete algebraic obstruction to the existence of closed smooth 4‐manifolds which are homotopy equivalent but not simple homotopy equivalent up to connected sum with .