Simple homotopy equivalence

Summary

A simple homotopy equivalence between finite CW complexes is a homotopy equivalence of a special kind. Up to homotopy, it is a finite sequence of elementary expansions and collapses, which add or remove a pair of cells in a trivial way. Not every homotopy equivalence is simple; the obstruction is the Whitehead torsion, which lies in the Whitehead group of the fundamental group. Simple homotopy type is finer than homotopy type and is the right notion for the s-cobordism theorem and for distinguishing lens spaces.