  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 with pairwise the same properties, and our examples
can be taken to be smooth for $n \geq 6$.
  Our key 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, and 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.
