Whitehead torsion

Summary

The Whitehead torsion $\tau(f)$ of a homotopy equivalence $f \colon K \to L$ of finite connected CW complexes is an element of the Whitehead group of $\pi_1(L)$. It vanishes exactly when $f$ is a simple homotopy equivalence. This means that $f$ is homotopic to a composite of finitely many elementary expansions and collapses of cells and their homotopy inverses. It is the invariant behind the s-cobordism theorem and it distinguishes homotopy equivalent spaces that are not simple homotopy equivalent, such as some lens spaces.