---
tag: F01D990B61
kind: definition
name: Simple homotopy equivalence
msc: [57Q10, 55P10]
defines: AR8SP73SXX
uses: [5W95QFHXPR, VBM8ZYTT2X, XZG6MTSWCN, 0Q0H13G8ZB, S1XK3HXHFX]
sources:
  - url: https://en.wikipedia.org/wiki/Simple_homotopy_theory
---

## Definition

Let $X$ be a finite [[CW complex|5W95QFHXPR]] and let $n \ge 1$.
Let $D^n$ be the [[closed unit ball|0Q0H13G8ZB]] of $\R^n$.
Its boundary sphere is the union of the closed lower hemisphere $D^{n-1}_-$ and the closed upper hemisphere $D^{n-1}_+$.
The two hemispheres meet in $S^{n-2}$.
Let $g \colon D^{n-1}_- \to X$ be a continuous map with $g(S^{n-2}) \subseteq X^{(n-2)}$ and $g(D^{n-1}_-) \subseteq X^{(n-1)}$, where $X^{(k)}$ is the [[$k$-skeleton|VBM8ZYTT2X]].
Let $Y = X \cup_g D^n$ be the space obtained by gluing $D^n$ to $X$ along $g$.
It is a CW complex with two new cells: the interior of $D^{n-1}_+$ and the interior of $D^n$.

1. The inclusion $X \hookrightarrow Y$ is an *elementary expansion*, and $Y$ is said to *collapse elementarily* onto $X$.
2. A *formal deformation* from $X$ to $Z$ is a finite sequence $X = X_0, X_1, \ldots, X_m = Z$ of finite CW complexes. Each step is an elementary expansion or an elementary collapse.
3. A cellular map $f \colon X \to Z$ is a *simple homotopy equivalence* if it is homotopic to the composite along a formal deformation from $X$ to $Z$. In this composite, an expansion contributes its inclusion and a collapse contributes a deformation retraction.

## Remarks

Let $X$ be a point and $n = 1$: gluing $D^1 = [-1, 1]$ to $X$ along its lower end point gives $Y = [-1, 1]$, with cells $\set{-1}$, $\set{1}$ and $(-1, 1)$.
So the inclusion of a point into an interval is an elementary expansion, and the constant map from the interval to the point is a simple homotopy equivalence.

The lens spaces $L(7, 1)$ and $L(7, 2)$ are [[homotopy equivalent|XZG6MTSWCN]], but no homotopy equivalence between them is simple.
Their Reidemeister torsions differ, and Reidemeister torsion is invariant under formal deformations.

Every simple homotopy equivalence is a homotopy equivalence.
Let $f \colon X \to Z$ be a homotopy equivalence of finite CW complexes.
Then $f$ is simple if and only if its Whitehead torsion $\tau(f)$ vanishes in the Whitehead group of the [[fundamental group|S1XK3HXHFX]] of $Z$.
In particular, if $Z$ is simply connected, every homotopy equivalence $X \to Z$ is simple, since the Whitehead group of the trivial group is zero.
By a theorem of Thomas Chapman, every homeomorphism between finite CW complexes is a simple homotopy equivalence.