Homotopic maps

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Definition

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

• there exists a homotopy, $H:X\times I\rightarrow Y$, such that $H_0=f$ and $H_1=g$ - here $I:=[0,1]\subset\mathbb{R}$ denotes the unit interval.

  (Recall for $t\in I$ that $H_t:X\rightarrow Y$ (which denotes a stage of the homotopy) is given by $H_t:x\mapsto H(x,t)$)

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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 $H:X\times I\rightarrow Y$
  • The stages of a homotopy are continuous
• Homotopy is an equivalence relation
• Path-homotopy
• Fundamental group