Homotopic maps

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Needs fleshing out, more references. Made a bit of a mess out of it, but I'll leave tidying up until later!


Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces, let [ilmath]f,g:X\rightarrow Y[/ilmath] be continuous maps and let [ilmath]A\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath].

  • We say "[ilmath]f[/ilmath] is homotopic to [ilmath]g[/ilmath] (relative to [ilmath]A[/ilmath])" if there exists a homotopy [ilmath](\text{rel }A)[/ilmath][Note 1] whose initial stage is [ilmath]f[/ilmath] and whose final stage is [ilmath]g[/ilmath].
    • This is written: [ilmath]f\simeq g\ (\text{rel}\ A)[/ilmath]
      • or simply [ilmath]f\simeq g[/ilmath] if [ilmath]A=\emptyset[/ilmath]
    • If [ilmath]A=\emptyset[/ilmath] (and we write [ilmath]f\simeq g[/ilmath]) we may say that [ilmath]f[/ilmath] and [ilmath]g[/ilmath] are freely homotopic
  • The homotopy [ilmath](\text{rel }A)[/ilmath] that exists if [ilmath]f\simeq g\ (\text{rel }A)[/ilmath], say [ilmath]F:X\times I\rightarrow Y[/ilmath], with [ilmath]\forall x\in X[(F(x,0)=f(x))\wedge(F(x,1)=g(x))][/ilmath] and [ilmath]\forall a\in A\forall t\in I[F(a,t)=f(a)=g(a)][/ilmath], is called a homotopy of maps
Explicit definition:

We say [ilmath]f\simeq g\ (\text{rel }A)[/ilmath] (or [ilmath]f\simeq g[/ilmath] if [ilmath]A=\emptyset[/ilmath]) if:

  • There exists a continuous map, [ilmath]F:X\times I\rightarrow Y[/ilmath] (a homotopy) such that:
    1. [ilmath]\forall x\in X[F(x,0)=f(x)][/ilmath] - the initial stage of the homotopy is [ilmath]f[/ilmath]
    2. [ilmath]\forall x\in X[F(x,1)=g(x)][/ilmath] - the final stage of the homotopy is [ilmath]g[/ilmath]
    3. [ilmath]\forall a\in A\forall s,t\in I[F(a,s)=F(a,t)][/ilmath][Note 2] - or equivalently - [ilmath]\forall a\in A\forall t\in I[F(a,t)=f(x)=g(x)][/ilmath] - the homotopy is fixed on [ilmath]A[/ilmath]

We can use this to define a relation on continuous maps:

  • If [ilmath]f\simeq g\ (\text{rel }A)[/ilmath] then we consider [ilmath]f[/ilmath] and [ilmath]g[/ilmath] related and say "[ilmath]f[/ilmath] is homotopic to [ilmath]g[/ilmath] ([ilmath]\text{rel }A[/ilmath])"

Claim: this is an equivalence relation (see: the relation of maps being homotopic is an equivalence relation)


  1. Recall a homotopy (relative to [ilmath]A[/ilmath]) is a continuous map, [ilmath]F:X\times I\rightarrow Y[/ilmath] (where [ilmath]I:=[0,1]\subset\mathbb{R}[/ilmath] - the unit interval) such that:
    • [ilmath]\forall a\in A\forall s,t\in I[F(a,t)=F(a,s)][/ilmath]
  2. Note that if [ilmath]A=\emptyset[/ilmath] then this represents no condition/constraint on [ilmath]F[/ilmath], as are not any [ilmath]a\in A[/ilmath] for this to be true on!




Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces. Let [ilmath]f,g:X\rightarrow Y[/ilmath] be continuous maps. The maps [ilmath]f[/ilmath] and [ilmath]h[/ilmath] are said to be homotopic[1] if:

  • there exists a homotopy, [ilmath]H:X\times I\rightarrow Y[/ilmath], such that [ilmath]H_0=f[/ilmath] and [ilmath]H_1=g[/ilmath] - here [ilmath]I:=[0,1]\subset\mathbb{R}[/ilmath] denotes the unit interval.
    (Recall for [ilmath]t\in I[/ilmath] that [ilmath]H_t:X\rightarrow Y[/ilmath] (which denotes a stage of the homotopy) is given by [ilmath]H_t:x\mapsto H(x,t)[/ilmath])

TODO: Mention free-homotopy, warn against using null (as that term is used for loops, mention relative homotopy

See also


  1. Introduction to Topological Manifolds - John M. Lee

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