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
[hide]Definition
Given topological spaces (X,J) and (Y,K), and any set A∈P(X)[Note 1] a homotopy (relative to A) is any continuous function:
- H:X×I→Y (where I:=[0,1]⊂R) such that:
- ∀s,t∈I ∀a∈A[H(a,s)=H(a,t)][Note 2]
If A=∅[Note 2] then we say H is a free homotopy (or just a homotopy).
If A≠∅ then we speak of a homotopy rel A or homotopy relative to A.
Stages of a homotopy
For a homotopy, H:X×I→Y (rel A), a stage of the homotopy H is a map:
- ht:X→Y for some t∈I given by ht:x↦H(x,t)
The family of maps, {ht:X→Y}t∈I, are collectively called the stages of a homotopy
Homotopy of maps
Notes
- Jump up ↑ Recall P(X) denotes the power set of X - the set containing all subsets of X; A⊆X⟺A∈P(X).
- ↑ Jump up to: 2.0 2.1 Note that if A=∅ then ∀s,t∈I ∀a∈∅[H(a,s)=H(a,t)] is trivially satisfied; it represents no condition. As there is no a∈∅ we never require H(a,s)=H(a,t).
References
OLD PAGE
Definition
Given two topological spaces, (X,J) and (Y,K) then a homotopy of maps (from X to Y) is a continuous function: F:X×I→Y (where I denotes the unit interval, I:=[0,1]⊂R). Note:
- The stages of the homotopy, F, are a family of functions, {ft:X→Y | t∈[0,1]} such that ft:x→F(x,t). The stages of a homotopy are continuous.
- f0 and f1 are examples of stages, and are often called the initial stage of the homotopy and final stage of the homotopy respectively.
Two (continuous) functions, g,h:X→Y are said to be homotopic if there exists a homotopy such that f0=g and f1=h
Notes
- Jump up ↑ 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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