Homotopy class

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The relation of paths being end-point-preserving homotopic is an Equivalence relation[1]

That is [ilmath]\alpha\simeq\beta\text{ rel}\{0,1\} [/ilmath] where [ilmath]\alpha[/ilmath] and [ilmath]\beta[/ilmath] are paths from [ilmath]a[/ilmath] to [ilmath]b[/ilmath] (which are not necessarily distinct as it may be a loop) is an equivalence relation, which is to say:

  • Reflexive: [ilmath]\alpha\simeq\alpha\text{ rel}\{0,1\} [/ilmath]
  • Symmetric: [ilmath]\alpha\simeq\beta\text{ rel}\{0,1\}\implies \beta\simeq\alpha\text{ rel}\{0,1\} [/ilmath]
  • Transitive: [ilmath]\alpha\simeq\beta\text{ rel}\{0,1\}\wedge\beta\simeq\gamma\text{ rel}\{0,1\}\implies \alpha\simeq\gamma\text{ rel}\{0,1\} [/ilmath]

The equivalence class of [ilmath]\alpha[/ilmath] is denoted (as is usual) by [ilmath][\alpha][/ilmath]

Important properties

[ilmath]\alpha,\ \beta[/ilmath] and [ilmath]\gamma[/ilmath] denote paths

  • For a continuous map [ilmath]p:[0,1]\rightarrow[0,1][/ilmath] with [ilmath]p(0)=0[/ilmath] and [ilmath]p(1)=1[/ilmath] we have:
    [ilmath][\alpha\circ p]=[\alpha][/ilmath] - that is any reparametrisation of [ilmath]\alpha[/ilmath] is homotopic to [ilmath]\alpha[/ilmath]
  • [math][\alpha_1]=[\alpha_2]\wedge[\beta_1]=[\beta_2]\implies[\alpha_1\beta_1]=[\alpha_2\beta_2][/math]
    • This allows us to define multiplication
  • [math](\alpha\beta)\gamma\simeq\alpha(\beta\gamma)\text{ rel}\{0,1\}[/math] or [math]([\alpha][\beta])[\gamma]=[\alpha]([\beta][\gamma])[/math]
    • This allows us to define associativity
  • Where [ilmath]a[/ilmath] is the constant loop at [ilmath]a[/ilmath] (ie [ilmath]a(t)=a\ \forall t\in[0,1][/ilmath]) we have
    [math]a\alpha\simeq\alpha\simeq\alpha b\text{ rel}\{0,1\}[/math] or [math][a][\alpha]=[\alpha]=[\alpha][b][/math]
  • if [ilmath]\alpha^{-1} [/ilmath] is the reverse path of [ilmath]\alpha[/ilmath], literally [ilmath]\alpha^{-1}(t)=\alpha(1-t)[/ilmath] then
    • [ilmath][\alpha_0]=[\alpha_1]\implies[\alpha_0^{-1}]=[\alpha_1^{-1}][/ilmath]
      • we can now define the inverse, [ilmath][\alpha^{-1}]=[\alpha]^{-1}[/ilmath]
  • [ilmath][\alpha][\alpha]^{-1}=[a][/ilmath]

TODO: Proofs for all of these p117

See also


  1. Introduction to topology - Second Edition - Theodore W. Gamelin and Rober Everist Greene