----- > [!proposition]+ Proposition. ([[homotopy equivalence without explicit homotopy inverses]]) > Suppose $f: X \to Y$ is a [[continuous]] map between [[topological space|topological spaces]] $X$ and $Y$ for which there exists [[continuous]] maps $g,h:Y \to X$ such that $g \circ f$ is [[homotopy|homotopic]] to $\id_{X}$ and $f \circ h$ is [[homotopy|homotopic]] to $\id_{Y}$. Then $f$, $g$, and $h$ are all [[homotopy equivalent|homotopy equivalences]]. ^proposition > [!proof]+ Proof. ([[homotopy equivalence without explicit homotopy inverses]]) > We liberally employ [[continuous functions respect homotopy]]. We are given that $g \circ f \simeq \id_{X}$ and $f \circ h \simeq \id_{Y}$. >```tikz \usepackage{tikz} \usepackage{tikz-cd} \begin{document} \begin{tikzcd} X \arrow[r, "f" description] & Y \arrow[l, "g"', bend right] \arrow[l, "h", bend left] \end{tikzcd} \end{document} >``` A parsimonious path through this diagram is $\ell := g \circ f \circ h$. Using this we observe that $g$ is [[homotopy equivalent|homotopic]] to $h$ because $g \simeq g \circ (f \circ h) = (g \circ f) \circ h \simeq h$. Therefore, $f \circ h \simeq \id_{X}$ implies $f \circ g \simeq \id_{X}$ and therefore $f$ and $g$ are [[homotopy equivalent|homotopy inverses]]. Likewise reasoning to show $f$ and $h$ are [[homotopy equivalent|homotopy inverses]]. ^proof #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch02