Difference between revisions of "Homotopic maps"

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{{Stub page|grade=A*|msg=Needs fleshing out, more references. Made a bit of a mess out of it, but I'll leave tidying up until later!}}
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==Definition==
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Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]], let {{M|f,g:X\rightarrow Y}} be ''[[continuous]]'' [[map|maps]] and let {{M|A\in\mathcal{P}(X)}} be an arbitrary subset of {{M|X}}.
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* We say "''{{M|f}} is homotopic to {{M|g}} (relative to {{M|A}})''" if there exists a [[homotopy]] {{M|(\text{rel }A)}}<ref group="Note">Recall a [[homotopy]] (relative to {{M|A}}) is a [[continuous map]], {{M|F:X\times I\rightarrow Y}} (where {{M|1=I:=[0,1]\subset\mathbb{R} }} - the [[unit interval]]) such that:
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* {{M|1=\forall a\in A\forall s,t\in I[F(a,t)=F(a,s)]}}</ref> whose {{link|initial stage|homotopy}} is {{M|f}} and whose {{link|final stage|homotopy}} is {{M|g}}.
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** This is written: {{M|f\simeq g\ (\text{rel}\ A)}}
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*** or simply {{M|f\simeq g}} if {{M|1=A=\emptyset}}
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** If {{M|1=A=\emptyset}} (and we write {{M|f\simeq g}}) we may say that {{M|f}} and {{M|g}} are ''freely homotopic''
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* The [[homotopy]] {{M|(\text{rel }A)}} that exists if {{M|f\simeq g\ (\text{rel }A)}}, say {{M|F:X\times I\rightarrow Y}}, with {{M|1=\forall x\in X[(F(x,0)=f(x))\wedge(F(x,1)=g(x))]}} and {{M|1=\forall a\in A\forall t\in I[F(a,t)=f(a)=g(a)]}}, is called a ''homotopy of maps''
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{{Begin Notebox}}
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'''Explicit definition:'''<br/>{{Begin Notebox Content}}
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We say {{M|f\simeq g\ (\text{rel }A)}} (or {{M|f\simeq g}} if {{M|1=A=\emptyset}}) if:
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* There exists a ''[[continuous]]'' [[map]], {{M|F:X\times I\rightarrow Y}} (a ''[[homotopy]]'') such that:
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*# {{M|1=\forall x\in X[F(x,0)=f(x)]}} - the {{link|initial stage|homotopy}} of the homotopy is {{M|f}}
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*# {{M|1=\forall x\in X[F(x,1)=g(x)]}} - the {{link|final stage|homotopy}} of the homotopy is {{M|g}}
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*# {{M|1=\forall a\in A\forall s,t\in I[F(a,s)=F(a,t)]}}<ref group="Note">Note that if {{M|1=A=\emptyset}} then this represents no condition/constraint on {{M|F}}, as are not any {{M|a\in A}} for this to be true on!</ref> - or equivalently - {{M|1=\forall a\in A\forall t\in I[F(a,t)=f(x)=g(x)]}} - the homotopy is fixed on {{M|A}}{{End Notebox Content}}{{End Notebox}}
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We can use this to define a [[relation]] on continuous maps:
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* If {{M|f\simeq g\ (\text{rel }A)}} then we consider {{M|f}} and {{M|g}} related and say "{{M|f}} is ''homotopic to'' {{M|g}} ({{M|\text{rel }A}})"
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'''Claim: ''' this is an equivalence relation (see: [[the relation of maps being homotopic is an equivalence relation]])
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==Notes==
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<references group="Note"/>
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==References==
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<references/>
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{{Definition|Topology|Homotopy Theory}}
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=OLD PAGE=
 
==Definition==
 
==Definition==
 
Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]]. Let {{M|f,g:X\rightarrow Y}} be [[continuous maps]]. The [[map|maps]] {{M|f}} and {{M|h}} are said to be ''homotopic''{{rITTMJML}} if:
 
Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]]. Let {{M|f,g:X\rightarrow Y}} be [[continuous maps]]. The [[map|maps]] {{M|f}} and {{M|h}} are said to be ''homotopic''{{rITTMJML}} if:

Latest revision as of 13:26, 15 September 2016

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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 [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)

Notes

  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!

References

OLD PAGE

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[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

References

  1. ↑ Introduction to Topological Manifolds - John M. Lee

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