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Concept
Simple-homotopy equivalence
Summary
In 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,
A homotopy theory that studies simple-homotopy types is called simple homotopy theory.
Official source
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