Difference between revisions of "HomotopyPage"
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| + | : '''Note: ''' a homotopy is really a relation on continuous functions (see [[homotopic]]), however since any continuous map of the form {{M|F:X\times I\rightarrow Y}} is a homotopy, and has continuous initial and final stages, it also means that, automatically, it's initial and final stages are [[homotopic maps]]. So simply by exhibiting a continuous function, {{M|F:X\times I\rightarrow Y}}, we get homotopic maps. So a homotopy warrants a definition, even if it is useless by itself. | ||
| + | __TOC__ | ||
==Definition== | ==Definition== | ||
Let {{Top.|X|J}} and {{Top.|Y|K}} be {{plural|topological space|s}} and let {{M|A\in\mathcal{P}(X)}} be an arbitrary subset of {{M|X}}. A ''homotopy, relative to {{M|A}}'' is, in its purest form, is any ''[[continuous]]'' [[map]]: | Let {{Top.|X|J}} and {{Top.|Y|K}} be {{plural|topological space|s}} and let {{M|A\in\mathcal{P}(X)}} be an arbitrary subset of {{M|X}}. A ''homotopy, relative to {{M|A}}'' is, in its purest form, is any ''[[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 group="Note">Note that if {{M|1=A=\emptyset}} then there is no {{M|a\in A}} and this represents no condition</ref> | + | * {{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 group="Note">Note that if {{M|1=A=\emptyset}} then there is no {{M|a\in A}} and this represents no condition</ref> - the homotopy is fixed on {{M|A}}. | ||
| + | There is some terminology used depending on whether or not {{M|1=A=\emptyset}}: | ||
| + | # {{M|1=A=\emptyset}} then we call {{M|F}} a ''free homotopy'' or just ''homotopy''. If however | ||
| + | # {{M|1=A\ne\emptyset}} then we speak of a ''homotopy relative to {{M|A}}'' or {{M|F\ (\text{rel}\ A)}} | ||
===Stages of the homotopy=== | ===Stages of the homotopy=== | ||
Let {{M|t\in I}} be given, and {{M|H:X\times I\rightarrow Y}} be a homotopy as defined above. {{M|h_t:X\rightarrow Y}} denotes a ''stage of the homotopy'' and is defined as follows: | Let {{M|t\in I}} be given, and {{M|H:X\times I\rightarrow Y}} be a homotopy as defined above. {{M|h_t:X\rightarrow Y}} denotes a ''stage of the homotopy'' and is defined as follows: | ||
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==References== | ==References== | ||
<references/> | <references/> | ||
| − | {{Definition|Topology}} | + | {{Definition|Topology|Homotopy Theory}} |
Latest revision as of 12:53, 15 September 2016
- Note: a homotopy is really a relation on continuous functions (see homotopic), however since any continuous map of the form [ilmath]F:X\times I\rightarrow Y[/ilmath] is a homotopy, and has continuous initial and final stages, it also means that, automatically, it's initial and final stages are homotopic maps. So simply by exhibiting a continuous function, [ilmath]F:X\times I\rightarrow Y[/ilmath], we get homotopic maps. So a homotopy warrants a definition, even if it is useless by itself.
Definition
Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces and let [ilmath]A\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath]. A homotopy, relative to [ilmath]A[/ilmath] is, in its purest form, is any 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][Note 1] - the homotopy is fixed on [ilmath]A[/ilmath].
There is some terminology used depending on whether or not [ilmath]A=\emptyset[/ilmath]:
- [ilmath]A=\emptyset[/ilmath] then we call [ilmath]F[/ilmath] a free homotopy or just homotopy. If however
- [ilmath]A\ne\emptyset[/ilmath] then we speak of a homotopy relative to [ilmath]A[/ilmath] or [ilmath]F\ (\text{rel}\ A)[/ilmath]
Stages of the homotopy
Let [ilmath]t\in I[/ilmath] be given, and [ilmath]H:X\times I\rightarrow Y[/ilmath] be a homotopy as defined above. [ilmath]h_t:X\rightarrow Y[/ilmath] denotes a stage of the homotopy and is defined as follows:
- [ilmath]h_t:x\mapsto H(x,t)[/ilmath]
The family of stages, [ilmath]\{h_t:X\rightarrow Y\}_{t\in I} [/ilmath], are collectively called the stages of the homotopy and
- [ilmath]h_0:X\rightarrow Y[/ilmath] defined by [ilmath]h_0:x\mapsto H(x,0)[/ilmath] is the initial stage of the homotopy.
- [ilmath]h_1:X\rightarrow Y[/ilmath] defined by [ilmath]h_1:x\mapsto H(x,1)[/ilmath] is the final stage of the homotopy.
Note that the stages of a homotopy are continuous
Purpose
The point of a homotopy is to be a relation (in fact an equivalence relation) of (continuous) maps.
Homotopic maps
Notes
- ↑ Note that if [ilmath]A=\emptyset[/ilmath] then there is no [ilmath]a\in A[/ilmath] and this represents no condition
