Abstract

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. It consequently proved extremely useful in manifold topology and is behind the s-cobordism theorem which is the basis for the vast majority of manifold classification results in dimension at least 4. 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. 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.