Homotopy equivalence

Spaces $X$ and $Y$ are homotopy equivalent when there are continuous mapssuch that $g\circ f$ is homotopic to ${\mathrm{id}}_X$ and $f\circ g$ is homotopic to ${\mathrm{id}}_Y$.

In the Oxford course's convention, a homotopy retract is a retraction $r:X\to A$ such that the inclusion map $i:A\hookrightarrow X$ also satisfies $ir\simeq {\mathrm{id}}_X$. It gives a homotopy equivalence between $A$ and $X$. Unlike the deformation retraction convention used here, the homotopy need not fix $A$ at intermediate times. ^[ref]

A homotopy equivalence of a closed surface cannot carry a simple closed curve of nonzero integral homology class onto a contractible simple closed curve. Any map into the latter factors through a circle whose inclusion induces zero on first homology; the former class would be killed by an isomorphism. Such a carrying map can nevertheless exist: project a handle generator onto a circle and compose with the contractible embedded circle.

If $f:X\to Y$ has maps $g,h:Y\to X$ with $fg\simeq 1_Y$ and $hf\simeq 1_X$, then $h\simeq hfg\simeq g$, so $g$ is a two-sided homotopy inverse. More generally, if $fg$ and $hf$ are homotopy equivalences, composing $g$ and $h$ with their respective homotopy inverses supplies a right and a left homotopy inverse of $f$, reducing to the first case.

Two spaces have the same homotopy type when there is a homotopy equivalence between them. Their homology and homotopy groups are then isomorphic. In Morse theory, attaching a cell records a change in the homotopy type of a sublevel set.

If $g\circ f\simeq {\mathrm{id}}_X$ and $f\circ g\simeq {\mathrm{id}}_Y$, then $g$ is a homotopy inverse of $f$ and conversely.