Difference between revisions of "Notes:Homotopy terminology"
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| + | __TOC__ | ||
| + | ==Plan== | ||
| + | # Homotopy - a thing, happens to be a relation on its terminal stages | ||
| + | # Homotopic - redirect to "Homotopic maps" - an equivalence relation on maps | ||
| + | # Homotopic paths - special case of homotopic maps | ||
| + | =OLD PAGE= | ||
==Homotopy== | ==Homotopy== | ||
Homotopy is a [[continuous map]], {{M|F:X\times I\rightarrow Y}} where {{M|I}} denotes the [[unit interval]], {{M|[0,1]\subseteq\mathbb{R} }}{{rATHHRMS}}.<br/> | Homotopy is a [[continuous map]], {{M|F:X\times I\rightarrow Y}} where {{M|I}} denotes the [[unit interval]], {{M|[0,1]\subseteq\mathbb{R} }}{{rATHHRMS}}.<br/> | ||
Revision as of 12:46, 14 September 2016
Contents
Plan
- Homotopy - a thing, happens to be a relation on its terminal stages
- Homotopic - redirect to "Homotopic maps" - an equivalence relation on maps
- Homotopic paths - special case of homotopic maps
OLD PAGE
Homotopy
Homotopy is a continuous map, [ilmath]F:X\times I\rightarrow Y[/ilmath] where [ilmath]I[/ilmath] denotes the unit interval, [ilmath][0,1]\subseteq\mathbb{R} [/ilmath][1].
Here [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] are topological spaces
Homotopic maps
If [ilmath]f,g:X\rightarrow Y[/ilmath] are continuous maps, we say that "[ilmath]f[/ilmath] is homotopic to [ilmath]g[/ilmath]" if[2]:
- There is a homotopy, [ilmath]F:X\times I\rightarrow Y[/ilmath] such that [ilmath]F(x,0)=f(x)[/ilmath] and [ilmath]F(x,1)=g(x)[/ilmath] ([ilmath]\forall x\in X[/ilmath])
Homotopic relative to [ilmath]A[/ilmath]
If [ilmath]f,g:X\rightarrow Y[/ilmath] are continuous maps and [ilmath]A\in\mathcal{P}(X)[/ilmath], we say that "[ilmath]f[/ilmath] is homotopic to [ilmath]g[/ilmath] relative to [ilmath]A[/ilmath]" if[3]:
- There is a homotopy, [ilmath]F:X\times I\rightarrow Y[/ilmath] such that [ilmath]F(x,0)=f(x)[/ilmath] and [ilmath]F(x,1)=g(x)[/ilmath] AND
- [ilmath]F(a,t)=f(a)=g(a)[/ilmath] for all [ilmath]t\in I[/ilmath] and [ilmath]\forall a\in A[/ilmath]
Homotopic paths
This is a special case, here we are dealing with [ilmath]A:=\{0,1\}[/ilmath] and [ilmath]F:I\times I\rightarrow X[/ilmath], the maps we are building a homotopy between are of the form:
- [ilmath]\alpha:I\rightarrow X[/ilmath]
And we say:
- [ilmath]\alpha_1[/ilmath] is homotopic to [ilmath]\alpha_2[/ilmath] (rel [ilmath]\{0,1\} [/ilmath]) if there is a homotopy between them.
