Two CW-complexes are simple homotopy equivalent if they are related by a sequence of collapses and expansions of cells. It implies homotopy equivalent as is implied by homeomorphic. This notion proved extremely useful in manifold topology and is central to the classification of non-simply connected manifolds up to homeomorphism. I will 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. The examples are constructed using surgery theory and the s-cobordism theorem, and are distinguished using methods from algebraic number theory and algebraic K-theory. I will also discuss a number of new directions including progress on classifying the possible fundamental groups for which examples exist. This is joint work with Csaba Nagy and Mark Powell.
