Homotopy and a normal homotopy between maps

[latentcorpse]

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 \alpha(0), \alpha(1) \in X of a path \alpha : I \rightarrow X 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 \alpha_0 , \alpha_1 : I \rightarrow X with the same endpoints \alpha_0(0) = \alpha_1(0) , \alpha_0 (1) = \alpha_1(1) \in X
is a collection of paths h_t : I \rightarrow X ( t \in [0,1]) witht he same endpoints h_t(0)=\alpha_0(0)=\alpha_1(0) , h_t(1) = \alpha_0(1) = \alpha_1(1)
such that h_0 = \alpha_0 , h_1 = \alpha_1 and such that the function
h : I \times I \rightarrow X ; (s,t) \mapsto h_t(s) 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 h_0 = \alpha_0 , h_1 = \alpha_1?
thanks.

 

[mrbohn1]

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 \{0,1\}.

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

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