Notes:Homotopy terminology

Caution:This page has been superceeded by Doctrine:Homotopy terminology and will be archived soon

Plan

1. Homotopy - a thing, happens to be a relation on its terminal stages
2. Homotopic - redirect to "Homotopic maps" - an equivalence relation on maps
3. 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].
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:

• 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:

• 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.