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!

Definition

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

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

[Expand] Explicit definition:

We say $f\simeq g\ (\text{rel }A)$ (or $f\simeq g$ if $A=\emptyset$) if:

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

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

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

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

OLD PAGE

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