Two CW-complexes are said to be simple homotopy equivalent if they are related by a sequence of collapses and expansions of cells. This notion interpolates between homeomorphism and homotopy in the sense that simple homotopy equivalent implies homotopy equivalent, and homeomorphic implies simple homotopy equivalent. The aim of this talk will be to present the first examples of two 4-manifolds which are homotopy equivalent but not simple homotopy equivalent, as well as in all higher even dimensions. This is joint work with Csaba Nagy and Mark Powell (arXiv:2312.00322). I will also discuss progress on the question of whether smooth 4-manifolds exist with these properties, through joint work with Daniel Kasprowski and Simona Veselá (arXiv:2405.06637). Finally, I will mention forthcoming work with Ian Hambleton in which we compare other closely related equivalence relations on 4-manifolds.

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