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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 (rel A)^{[Note 1]} whose initial stage is f and whose final stage is g.
- This is written: f≃g (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×I→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. $\forall a\in A\forall s,t\in I[F(a,s)=F(a,t)]$ ^{[Note 2]} - 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)

Notes

Jump up ↑
Recall a homotopy (relative to A) is a continuous map, F:X×I→Y (where I:=[0,1]⊂R - the unit interval) such that:
- $\forall a\in A\forall s,t\in I[F(a,t)=F(a,s)]$
Jump up ↑ Note that if $A=\emptyset$ then this represents no condition/constraint on $F$ , as are not any $a\in A$ for this to be true on!

References

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^[1] 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 $H_t:x\mapsto H(x,t)$ )

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

References

Jump up ↑ Introduction to Topological Manifolds - John M. Lee

Template:Homotopy theory navbox

[1] Jump up ↑ Recall a homotopy (relative to $A$ ) is a continuous map, $F:X\times I\rightarrow Y$ (where $I:=[0,1]\subset\mathbb{R}$ - the unit interval) such that:
$\forall a\in A\forall s,t\in I[F(a,t)=F(a,s)]$

[2] $\forall a\in A\forall s,t\in I[F(a,t)=F(a,s)]$

[2] Jump up ↑ Note that if $A=\emptyset$ then this represents no condition/constraint on $F$ , as are not any $a\in A$ for this to be true on!

[ITTMJML-3] Jump up ↑ Introduction to Topological Manifolds - John M. Lee

[Note 1]

[Note 2]

[1]

@@ Line 1: / Line 1: @@
-{{Stub page|grade=A*|msg=Needs fleshing out, more references}}
+{{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!}}
+==Definition==
+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}}.
+* 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:
+* {{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}}.
+** This is written: {{M|f\simeq g\ (\text{rel}\ A)}}
+*** or simply {{M|f\simeq g}} if {{M|1=A=\emptyset}}
+** If {{M|1=A=\emptyset}} (and we write {{M|f\simeq g}}) we may say that {{M|f}} and {{M|g}} are ''freely homotopic''
+* 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''
+{{Begin Notebox}}
+'''Explicit definition:'''<br/>{{Begin Notebox Content}}
+We say {{M|f\simeq g\ (\text{rel }A)}} (or {{M|f\simeq g}} if {{M|1=A=\emptyset}}) if:
+* There exists a ''[[continuous]]'' [[map]], {{M|F:X\times I\rightarrow Y}} (a ''[[homotopy]]'') such that:
+*# {{M|1=\forall x\in X[F(x,0)=f(x)]}} - the {{link|initial stage|homotopy}} of the homotopy is {{M|f}}
+*# {{M|1=\forall x\in X[F(x,1)=g(x)]}} - the {{link|final stage|homotopy}} of the homotopy is {{M|g}}
+*# {{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}}
+We can use this to define a [[relation]] on continuous maps:
+* 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}})"
+'''Claim: ''' this is an equivalence relation (see: [[the relation of maps being homotopic is an equivalence relation]])
+==Notes==
+<references group="Note"/>
+==References==
+<references/>
+{{Definition|Topology|Homotopy Theory}}
+=OLD PAGE=
 ==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:

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Latest revision as of 13:26, 15 September 2016

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