In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, particularly the area of topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, a simple-homotopy equivalence is a refinement of the concept of homotopy equivalence. Two CW-complexes are simple-homotopy equivalent if they are related by a sequence of collapses

Collapse (topology)

In topology, a branch of mathematics, collapse is a concept due to J. H. C. Whitehead.- Definition :Let K be a simplicial complex, and suppose that s is a simplex in K. We say that s has a free face t if t is a face of s and t has no other cofaces. We call a free pair...

 and expansions (inverses of collapses), and a homotopy equivalence is a simple homotopy equivalence if it is homotopic to such a map.

The obstruction to a homotopy equivalence being a simple homotopy equivalence is the Whitehead torsion

Whitehead torsion

In geometric topology, the obstruction to a homotopy equivalence f\colon X \to Y of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion \tau, which is an element in the Whitehead group Wh. These are named after the mathematician J. H. C...

,