Definition 2

Let X and Y be topological spaces. A homotopy class $\varphi \in$ [X; Y] is said to be invertible if there exists a homotopy class $\psi \in [Y; X]$ such that $\psi \circ \varphi =$ [Id$_X]$ and $\varphi \circ \psi =$ [Id$_Y]$. A continuous mapping is said to be a homotopy equivalence if its homotopy class is invertible.

Let $\varphi \in$ [X; Y] be an invertible homotopy class. There exists a unique homotopy class $\psi \in [Y; X]$ having the properties of definition 2. Let indeed $\psi ,\psi '$ be classes having these properties; one has

$\psi =\psi \circ$ [Id$_Y] =\psi \circ \varphi \circ \psi '=$ [Id$_X]\circ \psi '=\psi '$.

This unique class is called the inverse of the homotopy class $\varphi$ and is denoted $\varphi^{-1}$.

Let Z be a topological space. We have $\chi \circ \varphi \circ \varphi^{-1}=\chi$ for all $\chi \in$ [Y; Z] and $\theta \circ \varphi^{-1}\circ \varphi =\theta$ for all $\theta \in [X; Z]$. It follows that the mappings $\chi \mapsto \chi \circ \varphi$ of [Y; Z] into [X; Z] and $\theta \mapsto \theta \circ \varphi^{-1}$ of [X; Z] into [Y; Z] are bijections inverse to one another. Analogously, the mapping $\chi \mapsto \varphi \circ \chi$ of [Z; X] into [Z; Y] is bijective and its inverse bijection is the mapping $\theta \mapsto \varphi^{-1}\circ \theta$ of [Z; Y] into [Z; X].

Let X, Y, Z be topological spaces, $\varphi \in [X; Y]$ and $\psi \in [Y; Z]$ invertible homotopy classes. Then the class $\psi \circ \varphi$ is invertible, with inverse $\varphi^{-1}\circ \psi^{-1}$. Indeed, we have

$=\psi \circ$ [Id$_Y]\circ \psi^{-1}$

$=\psi \circ \psi^{-1}=$ [Id$_Z]$

and, analogously, $(\varphi^{-1}\circ \psi^{-1})\circ (\psi \circ \varphi ) =$ [Id$_X]$.

Let X, Y be topological spaces and let $f: X\rightarrow Y$ be a homotopy equivalence. Any continuous mapping $g: Y\rightarrow X$ whose class is the inverse of that of $f$ is called a reciprocal (or inverse) of $f$ up to homotopy. Such a mapping $g$ is a homotopy equivalence.

A homeomorphism $f$ is a homotopy equivalence, and $[f]^{-1}= [f^{-1}]$. The composite mapping of two homotopy equivalences is a homotopy equivalence. The relation "X and Y are topological spaces and there exists a homotopy equivalence of X into Y" is an equivalence relation.