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.