Stable equivalence relations on 4-manifolds

Kasprowski, Daniel, Nicholson, John and Veselá, Simona (2025) Stable equivalence relations on 4-manifolds. Proceedings of the London Mathematical Society. (doi:10.48550/arXiv.2405.06637). (In Press)

Record type: Article

Abstract

Kreck's modified surgery gives an approach to classifying smooth -manifolds up to stable diffeomorphism, i.e. 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 stable homeomorphism for homotopy 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 .

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