---
tag: AR8SP73SXX
kind: concept
name: Simple homotopy equivalence
msc: [57Q10, 55P10]
specializes: [XZG6MTSWCN]
sources:
  - url: https://en.wikipedia.org/wiki/Simple_homotopy_theory
---

## 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.

## Explanations

### L1 human

Imagine a shape made of cells, like a model built of balls and disks.
An elementary expansion glues on a new solid ball along half of its boundary.
This adds two cells but changes nothing up to deformation, since the ball can be squashed back flat.
An elementary collapse is the reverse move.
Two complexes are simple homotopy equivalent when one can pass from one to the other by finitely many such moves.
A solid square and a single point are simple homotopy equivalent: collapse the square onto an edge, then the edge onto a point.