Properties:: *[[Properties]]* Sufficiencies:: [[homotopy equivalence without explicit homotopy inverses]] Equivalences:: *[[Equivalences]]* ---- $X$ and $Y$ are [[topological space|topological spaces]]. > [!definition] Definition. ([[homotopy equivalent]]) Let $f:X \to Y$ and $g:Y \to X$ be [[continuous]]. Suppose >- $g \circ f:X \to X$ is [[homotopy|homotopic]] to $\id_{X}$ and >- $f \circ g: Y \to Y$ is [[homotopy|homotopic]] to $\id_{Y}$. > Then the maps $f,g$ are called **homotopy equivalences/inverses** and spaces $X$ and $Y$ are called **homotopy equivalent**. > The [[relation]] of homotopy equivalence is an [[equivalence relation]]. Two homotopy equivalent spaces are said to have the same **homotopy type**. \ Note that this is a relaxation of the notion of being [[homeomorphism]], as well as (in a different sense) the notion of a [[deformation retract]]. - as a [[category]] > [!justification] > > It is straightforward to show that the [[relation]] $\sim$ of homotopy equivalence is an [[equivalence relation]]. $X$ is obviously [[homeomorphism]] to itself, so reflexivity is satisfied. If $X$ is [[homotopy equivalent]] to $Y$ then $Y$ is homotopy equivalent to $X$ as a matter of definition, so symmetry is satisfied. > > i wrote how to do this in the book. > [!example] Examples. > - If $A \subset X$ is a [[deformation retract]] of $X$, then $A$ and $X$ are [[homotopy equivalent]]. For by definitions, $\iota_{A} \circ r \simeq \id_{X}$ and $r \circ \iota_{A}=\id_{A}$. *Actually, a flavor of the converse might be true...* > - [[complement of spheres is homotopy equivalent to another one]] ^example ---- #### ---- #### 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 > ```