Abstract: A homotopy equivalence between finite CW-complexes is called simple if it is homotopic to a composition of elementary collapses and expansions. Lens spaces provide famous examples of manifolds that are homotopy equivalent but not simple homotopy equivalent to each other, in all $\geq 3$ odd dimensions. However, no even-dimensional examples are known in the literature. We construct even-dimensional manifolds that are homotopy equivalent (in fact h- cobordant) but not simple homotopy equivalent to each other. More generally, we give a purely algebraic characterisation of groups G with the property that there exists a pair of manifolds with fundamental group G that are h-cobordant but not simple homotopy equivalent. This is joint work with Johnny Nicholson and Mark Powell.