Simple homotopy types of even dimensional manifolds

Abstract

Given a closed $n$-manifold, we consider the set of simple homotopy types of $n$-manifolds within its homotopy type, called its simple homotopy manifold set. We characterise it in terms of algebraic K-theory, the surgery obstruction map, and the homotopy automorphisms of the manifold. We use this to construct the first examples, for all $n \ge 4$ even, of closed $n$-manifolds that are homotopy equivalent but not simple homotopy equivalent. In fact, we construct infinite families of manifolds that are all homotopy equivalent but pairwise not simple homotopy equivalent, and our examples can be taken to be smooth for $n \geq 6$. Our examples are homotopy equivalent to the product of a circle and a lens space. We analyse the simple homotopy manifold sets of these manifolds, determining exactly when they are trivial, finite, or infinite, and investigating their asymptotic behaviour. The proofs involve integral representation theory and class numbers of cyclotomic fields. We also compare with the relation of $h$-cobordism, and produce similar detailed quantitative descriptions of the manifold sets that arise.