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  "body": "\n## Definition\n\nLet $X$ be a finite [[CW complex|5W95QFHXPR]] and let $n \\ge 1$.\nLet $D^n$ be the [[closed unit ball|0Q0H13G8ZB]] of $\\R^n$.\nIts boundary sphere is the union of the closed lower hemisphere $D^{n-1}_-$ and the closed upper hemisphere $D^{n-1}_+$.\nThe two hemispheres meet in $S^{n-2}$.\nLet $g \\colon D^{n-1}_- \\to X$ 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]].\nLet $Y = X \\cup_g D^n$ be the space obtained by gluing $D^n$ to $X$ along $g$.\nIt is a CW complex with two new cells: the interior of $D^{n-1}_+$ and the interior of $D^n$.\n\n1. The inclusion $X \\hookrightarrow Y$ is an *elementary expansion*, and $Y$ is said to *collapse elementarily* onto $X$.\n2. 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.\n3. 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.\n\n## Remarks\n\nLet $X$ 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 $(-1, 1)$.\nSo 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.\n\nThe lens spaces $L(7, 1)$ and $L(7, 2)$ are [[homotopy equivalent|XZG6MTSWCN]], but no homotopy equivalence between them is simple.\nTheir Reidemeister torsions differ, and Reidemeister torsion is invariant under formal deformations.\n\nEvery simple homotopy equivalence is a homotopy equivalence.\nLet $f \\colon X \\to Z$ be a homotopy equivalence of finite CW complexes.\nThen $f$ is simple if and only if its Whitehead torsion $\\tau(f)$ vanishes in the Whitehead group of the [[fundamental group|S1XK3HXHFX]] of $Z$.\nIn particular, if $Z$ is simply connected, every homotopy equivalence $X \\to Z$ is simple, since the Whitehead group of the trivial group is zero.\nBy a theorem of Thomas Chapman, every homeomorphism between finite CW complexes is a simple homotopy equivalence.\n",
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