Homotopy equivalent topological spaces

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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]


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])
      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!

See next



  1. Introduction to Topological Manifolds - John M. Lee