In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly the area of
topology, 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 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 In geometric topology, a field within mathematics, 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(f) which is an element in the Whitehead group \ope ...
,
A homotopy theory that studies simple-homotopy types is called
simple homotopy theory
In mathematics, simple homotopy theory is a homotopy theory (a branch of algebraic topology) that concerns with the simple-homotopy type of a space. It was originated by Whitehead in his 1950 paper "Simple homotopy type".
See also
*Whitehead tor ...
.
See also
*
Discrete Morse theory
References
*
Homotopy theory
Equivalence (mathematics)
{{topology-stub