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Homotopy and a normal homotopy between maps

  • #1
1,444
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I can't see a difference between a rel{0,1} homotopy and a normal homotopy between maps.

my notes say:
What does a rel{0,1} homotopy mean? Keeping the endpoints [itex] \alpha(0), \alpha(1) \in X[/itex] of a path [itex]\alpha : I \rightarrow X[/itex] fixed during a homotopy.

we do this anyway don't we?

and then more precisely they say:
A homotopy rel {0,1} of two paths [itex] \alpha_0 , \alpha_1 : I \rightarrow X[/itex] with the same endpoints [itex]\alpha_0(0) = \alpha_1(0) , \alpha_0 (1) = \alpha_1(1) \in X[/itex]
is a collection of paths [itex]h_t : I \rightarrow X ( t \in [0,1]) [/itex] witht he same endpoints [itex]h_t(0)=\alpha_0(0)=\alpha_1(0) , h_t(1) = \alpha_0(1) = \alpha_1(1)[/itex]
such that [itex]h_0 = \alpha_0 , h_1 = \alpha_1[/itex] and such that the function
[itex] h : I \times I \rightarrow X ; (s,t) \mapsto h_t(s)[/itex] is continuous.

I can't see much difference between this and the standard homotopy definition other than the fact that such a homotopy could only be defined for maps whose endpoints are the same to begin with.

also how does [itex]h_0 = \alpha_0 , h_1 = \alpha_1[/itex]???
thanks.
 

Answers and Replies

  • #2
97
0


If two paths are homotopic relative to a subspace, then it just means that they take the same values on that subspace. You are right in saying that the only distinction between this and a general homotopy is that the maps must agree on this subspace. In this case the subspace is [tex]\{0,1\}[/tex].

Saying two paths are homotopic does not necessarily imply that they have the same endpoints.

[tex]h_0=\alpha_0[/tex] just means that at [tex]t=0, h[/tex] agrees with [tex]\alpha_0[/tex] for all [tex]s.[/tex] Similarly, at [tex]t=1, h[/tex] agrees with [tex]\alpha_1[/tex] for all [tex]s.[/tex] Think of two paths, with the same endpoints, and infinitely many paths in between them....as [tex]t[/tex] increases the path we are on changes, and as [tex]s[/tex] increases our position on the path (how far along that path we are) changes.
 

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