Simple homotopy equivalence

Let $X$ and $Y$ be finite CW complexes. A map $f:X\to Y$ is a simple homotopy equivalence if it is homotopic to a finite composite $X=X_0\stackrel{f_1}{\to }{X}_1\stackrel{f_2}{\to }\cdots \stackrel{f_k}{\to }{X}_k=Y$ of maps between finite CW complexes in which each $f_i$ is either an elementary expansion inclusion, an elementary collapse map (Elementary expansions and collapses of finite CW complexes), or a cellular isomorphism. A cellular isomorphism here means a homeomorphism that carries the cell structure of its source isomorphically onto the cell structure of its target, so that it restricts to a homeomorphism between the interiors of corresponding cells; a general homeomorphism is not included by definition.

For a formal collapse $Y\searrow X$, an elementary collapse map means any retraction $r:Y\to X$ of its expansion inclusion $j:X\hookrightarrow Y$. Such maps exist: the characteristic ball strongly deformation retracts onto the complementary boundary disk, fixing that disk pointwise. This deformation descends through the attaching identifications to a strong deformation retraction of $Y$ onto $X$. If $r_0$ is its endpoint and $r$ is any retraction, composing this deformation with $r$ gives $r\simeq r_0$ relative to $X$. Thus $rj={\mathrm{i}\mathrm{d}}_X$ and $jr\simeq {\mathrm{i}\mathrm{d}}_Y$.

Every such composite is a homotopy equivalence (Homotopy equivalences, homotopy inverses and spaces of the same homotopy type): an expansion inclusion and its collapse map are homotopy inverses, and a cellular isomorphism is a homeomorphism. Consequently every simple homotopy equivalence is a homotopy equivalence.

For disconnected complexes the sequence respects the induced bijection on components: each operation is performed componentwise, and the composite maps the components of $X$ bijectively onto those of $Y$. The empty complex is allowed; the empty sequence exhibits the identity of a finite CW complex as a simple homotopy equivalence.

Homotopy of maps is transitive, so any map homotopic to a simple homotopy equivalence is again a simple homotopy equivalence. Concatenating two finite composites exhibits a composite of two simple homotopy equivalences as a simple homotopy equivalence as well.