Difference between revisions of "Homotopy"
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| − | {{Stub page|grade=A}} | + | {{Stub page|msg=This was to be swapped or merged with [[homotopyPage]] - don't forget, spotted more than a year later! [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 19:31, 26 November 2017 (UTC) |grade=A}} |
{{Requires references|grade=A}} | {{Requires references|grade=A}} | ||
__TOC__ | __TOC__ | ||
| + | ==Definition== | ||
| + | Given [[topological spaces]] {{Top.|X|J}} and {{Top.|Y|K}}, and any [[set]] {{M|A\in\mathcal{P}(X)}}<ref group="Note">Recall {{M|\mathcal{P}(X)}} denotes the [[power set]] of {{M|X}} - the set containing all subsets of {{M|X}}; {{M|A\subseteq X\iff A\in\mathcal{P}(X)}}.</ref> a ''homotopy (relative to {{M|A}})'' is any [[continuous function]]: | ||
| + | * {{M|H:X\times I\rightarrow Y}} (where {{M|1=I:=[0,1]\subset\mathbb{R} }}) such that: | ||
| + | ** {{M|1=\forall s,t\in I\ \forall a\in A[H(a,s)=H(a,t)]}}<ref group="Note" name="Emptysetcase">Note that if {{M|1=A=\emptyset}} then {{M|1=\forall s,t\in I\ \forall a\in\emptyset[H(a,s)=H(a,t)]}} is trivially satisfied; it represents no condition. As there is no {{M|a\in\emptyset}} we never require {{M|1=H(a,s)=H(a,t)}}.</ref> | ||
| + | If {{M|1=A=\emptyset}}<ref group="Note" name="Emptysetcase"/> then we say {{M|H}} is a ''free homotopy'' (or just a ''homotopy'').<br/> | ||
| + | If {{M|A\neq \emptyset}} then we speak of a ''homotopy rel {{M|A}}'' or ''homotopy relative to {{M|A}}''. | ||
| + | ===Stages of a homotopy=== | ||
| + | For a homotopy, {{M|H:X\times I\rightarrow Y\ (\text{rel }A)}}, a ''stage of the homotopy {{M|H}}'' is a map: | ||
| + | * {{M|h_t:X\rightarrow Y}} for some {{M|t\in I}} given by {{M|h_t:x\mapsto H(x,t)}} | ||
| + | The family of maps, {{M|\{h_t:X\rightarrow Y\}_{t\in I} }}, are collectively called the ''stages of a homotopy'' | ||
| + | * '''Note: ''' [[the stages of a homotopy are continuous]] | ||
| + | ==Homotopy of maps== | ||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
| + | ==References== | ||
| + | <references/> | ||
| + | |||
| + | =OLD PAGE= | ||
==Definition== | ==Definition== | ||
Given two [[topological spaces]], {{Top.|X|J}} and {{Top.|Y|K}} then a ''homotopy of maps (from {{M|X}} to {{M|Y}})'' is a ''[[continuous]]'' [[function]]: {{M|F:X\times I\rightarrow Y}} (where {{M|I}} denotes the [[unit interval]], {{M|1=I:=[0,1]\subset\mathbb{R} }}). Note: | Given two [[topological spaces]], {{Top.|X|J}} and {{Top.|Y|K}} then a ''homotopy of maps (from {{M|X}} to {{M|Y}})'' is a ''[[continuous]]'' [[function]]: {{M|F:X\times I\rightarrow Y}} (where {{M|I}} denotes the [[unit interval]], {{M|1=I:=[0,1]\subset\mathbb{R} }}). Note: | ||
| − | * The ''stages of the homotopy, {{M|F}},'' are a family of functions, {{M|\{ f_t:X\rightarrow Y\ \vert\ t\in[0,1]\} }} such that {{M|f_t:x\rightarrow F(x,t)}}. [[The stages of a homotopy are continuous]]. | + | * The '''''stages of the homotopy, {{M|F}},''''' are a family of functions, {{M|\{ f_t:X\rightarrow Y\ \vert\ t\in[0,1]\} }} such that {{M|f_t:x\rightarrow F(x,t)}}. [[The stages of a homotopy are continuous]]. |
** {{M|f_0}} and {{M|f_1}} are examples of stages, and are often called the ''initial stage of the homotopy'' and ''final stage of the homotopy'' respectively. | ** {{M|f_0}} and {{M|f_1}} are examples of stages, and are often called the ''initial stage of the homotopy'' and ''final stage of the homotopy'' respectively. | ||
Two ([[continuous]]) functions, {{M|g,h:X\rightarrow Y}} are said to be ''homotopic'' if there exists a homotopy such that {{M|1=f_0=g}} and {{M|1=f_1=h}} | Two ([[continuous]]) functions, {{M|g,h:X\rightarrow Y}} are said to be ''homotopic'' if there exists a homotopy such that {{M|1=f_0=g}} and {{M|1=f_1=h}} | ||
Latest revision as of 19:31, 26 November 2017
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This was to be swapped or merged with homotopyPage - don't forget, spotted more than a year later! Alec (talk) 19:31, 26 November 2017 (UTC)
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Contents
Definition
Given topological spaces [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath], and any set [ilmath]A\in\mathcal{P}(X)[/ilmath][Note 1] a homotopy (relative to [ilmath]A[/ilmath]) is any continuous function:
- [ilmath]H:X\times I\rightarrow Y[/ilmath] (where [ilmath]I:=[0,1]\subset\mathbb{R}[/ilmath]) such that:
- [ilmath]\forall s,t\in I\ \forall a\in A[H(a,s)=H(a,t)][/ilmath][Note 2]
If [ilmath]A=\emptyset[/ilmath][Note 2] then we say [ilmath]H[/ilmath] is a free homotopy (or just a homotopy).
If [ilmath]A\neq \emptyset[/ilmath] then we speak of a homotopy rel [ilmath]A[/ilmath] or homotopy relative to [ilmath]A[/ilmath].
Stages of a homotopy
For a homotopy, [ilmath]H:X\times I\rightarrow Y\ (\text{rel }A)[/ilmath], a stage of the homotopy [ilmath]H[/ilmath] is a map:
- [ilmath]h_t:X\rightarrow Y[/ilmath] for some [ilmath]t\in I[/ilmath] given by [ilmath]h_t:x\mapsto H(x,t)[/ilmath]
The family of maps, [ilmath]\{h_t:X\rightarrow Y\}_{t\in I} [/ilmath], are collectively called the stages of a homotopy
Homotopy of maps
Notes
- ↑ Recall [ilmath]\mathcal{P}(X)[/ilmath] denotes the power set of [ilmath]X[/ilmath] - the set containing all subsets of [ilmath]X[/ilmath]; [ilmath]A\subseteq X\iff A\in\mathcal{P}(X)[/ilmath].
- ↑ 2.0 2.1 Note that if [ilmath]A=\emptyset[/ilmath] then [ilmath]\forall s,t\in I\ \forall a\in\emptyset[H(a,s)=H(a,t)][/ilmath] is trivially satisfied; it represents no condition. As there is no [ilmath]a\in\emptyset[/ilmath] we never require [ilmath]H(a,s)=H(a,t)[/ilmath].
References
OLD PAGE
Definition
Given two topological spaces, [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] then a homotopy of maps (from [ilmath]X[/ilmath] to [ilmath]Y[/ilmath]) is a continuous function: [ilmath]F:X\times I\rightarrow Y[/ilmath] (where [ilmath]I[/ilmath] denotes the unit interval, [ilmath]I:=[0,1]\subset\mathbb{R}[/ilmath]). Note:
- The stages of the homotopy, [ilmath]F[/ilmath], are a family of functions, [ilmath]\{ f_t:X\rightarrow Y\ \vert\ t\in[0,1]\} [/ilmath] such that [ilmath]f_t:x\rightarrow F(x,t)[/ilmath]. The stages of a homotopy are continuous.
- [ilmath]f_0[/ilmath] and [ilmath]f_1[/ilmath] are examples of stages, and are often called the initial stage of the homotopy and final stage of the homotopy respectively.
Two (continuous) functions, [ilmath]g,h:X\rightarrow Y[/ilmath] are said to be homotopic if there exists a homotopy such that [ilmath]f_0=g[/ilmath] and [ilmath]f_1=h[/ilmath]
Notes
- ↑ Do not shorten this to "homotopy equivalence" as homotopy equivalence of spaces is something very different
References
Template:Algebraic topology navbox
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