Homotopy equivalence and simple homotopy equivalence of manifolds

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.