# Homotopy equivalent topological spaces

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## Definition

Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces, we say [ilmath]X[/ilmath] is *homotopy equivalent* to [ilmath]Y[/ilmath], or [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] have the same *homotopy type*, written [ilmath]X\simeq Y[/ilmath], if^{[1]}:

- [ilmath]\exists f\in[/ilmath][ilmath]C(X,Y)[/ilmath][ilmath]\exists g\in C(Y,X)\big[(g\circ f\simeq [/ilmath][ilmath]\text{Id}_X[/ilmath][ilmath])\wedge(g\circ f\simeq \text{Id}_Y)\big][/ilmath]
- Here [ilmath]C(X,Y)[/ilmath] denotes the set of continuous maps from [ilmath]X[/ilmath] and [ilmath]f\simeq g[/ilmath] denotes the relation of homotopy of maps - that is in this case freely homotopic
- In words, there exist two continuous maps, [ilmath]f:X\rightarrow Y[/ilmath] and [ilmath]g:Y\rightarrow X[/ilmath] such that [ilmath](g\circ f):X\rightarrow X[/ilmath] is freely homotopic to [ilmath]\text{Id}_X:X\rightarrow X[/ilmath] (the identity map on [ilmath]X[/ilmath]) with [ilmath]\text{Id}_X:x\mapsto x[/ilmath] and [ilmath](f\circ g):Y\rightarrow Y[/ilmath] is again freely homotopic to [ilmath]\text{Id}_Y:Y\rightarrow Y[/ilmath] by [ilmath]\text{Id}_Y:y\mapsto y[/ilmath]

### Terminology

Let [ilmath]f:X\rightarrow Y[/ilmath] be a continuous map (so [ilmath]f\in C(X,Y)[/ilmath] in other words) and let [ilmath]g:Y\rightarrow X[/ilmath] be another continuous map (so [ilmath]g\in C(Y,X)[/ilmath], as before), then:

- if [ilmath](g\circ f)\simeq \text{Id}_X[/ilmath] and [ilmath](f\circ g)\simeq\text{Id}_Y[/ilmath] (so [ilmath]X\simeq Y[/ilmath], as is the topic of this page) then we may say:
- [ilmath]g[/ilmath] is a
*homotopy inverse*for [ilmath]f[/ilmath] - [ilmath]f[/ilmath] is a
*homotopy equivalence*(as it has a*homotopy inverse*, namely [ilmath]g[/ilmath])- Note also that if [ilmath]X\simeq Y[/ilmath] with [ilmath]f[/ilmath] and [ilmath]g[/ilmath] then [ilmath]Y\simeq X[/ilmath] with [ilmath]g[/ilmath] and [ilmath]f[/ilmath] as the maps, leading to:
- [ilmath]f[/ilmath] is a
*homotopy inverse*for [ilmath]g[/ilmath] and - [ilmath]g[/ilmath] is a
*homotopy equivalence*(as it has a*homotopy inverse*, namely [ilmath]f[/ilmath])

- [ilmath]f[/ilmath] is a

- We will later see that homotopy equivalence of topological spaces is an equivalence relation, so if [ilmath]X\simeq Y[/ilmath] then [ilmath]Y\simeq X[/ilmath], as we've basically just shown, the
*symmetric property of an equivalence relation*is easy to see!

- Note also that if [ilmath]X\simeq Y[/ilmath] with [ilmath]f[/ilmath] and [ilmath]g[/ilmath] then [ilmath]Y\simeq X[/ilmath] with [ilmath]g[/ilmath] and [ilmath]f[/ilmath] as the maps, leading to:

- [ilmath]g[/ilmath] is a

## See next

- Homotopy equivalence of topological spaces is an equivalence relation - the relation of [ilmath]X\simeq Y[/ilmath] is an equivalence relation
- A deformation retract induces a homotopy equivalence
- If [ilmath]A[/ilmath] (as a topological subspace of [ilmath](X,\mathcal{ J })[/ilmath]) is a deformation retract of [ilmath]X[/ilmath] then [ilmath]X\simeq A[/ilmath]

- Homotopy invariance of the fundamental group - homotopically equivalent spaces have isomorphic fundamental groups
- If [ilmath]f:X\rightarrow Y[/ilmath] is a homotopy equivalence (so [ilmath]X\simeq Y[/ilmath] through [ilmath]f[/ilmath] and some other map which would be its
*homotopy inverse*) then:- [ilmath]\forall p\in X[\pi_1(X,p)\cong_{f_*}\pi_1(Y,f(p))][/ilmath] where [ilmath]\cong[/ilmath] denotes isomorphism of groups and [ilmath]f_*:\pi_1(X,p)\rightarrow\pi_1(Y,f(p))[/ilmath] the fundamental group homomorphism induced by a continuous map

- If [ilmath]f:X\rightarrow Y[/ilmath] is a homotopy equivalence (so [ilmath]X\simeq Y[/ilmath] through [ilmath]f[/ilmath] and some other map which would be its