Homotopic maps

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Definition

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 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])

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

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See also

• Homotopy - any continuous map of the form [ilmath]H:X\times I\rightarrow Y[/ilmath]
  • The stages of a homotopy are continuous
• Homotopy is an equivalence relation
• Path-homotopy
• Fundamental group