Using Straight-Line Homotopies to Construct Free Homotopies

[littleHilbert]

Hello,

Here is a short lemma:

A path-connected space X is simply-connected iff any two loops in X are free homotopic.

My question is whether it is allowed to use a straight-line homotopy straight away in order to construct a free homotopy? For example, let u and v be two loops and w is a curve from point a to point b. Then:


$\begin{equation} H_f(t,s):= \begin{cases} (1-3s)u(t)+3sa,\ \text{for 0\leqslant{}s\leqslant\frac{1}{3}}; \\ w(3s-1),\ \text{for \frac{1}{3}\leqslant{}s\leqslant\frac{2}{3}}; \\ (3-3s)b+(3s-2)v(t),\ \text{for \frac{2}{3}\leqslant{}s\leqslant1}. \end{cases} \end{equation}$
(there is an error in the latex-output: "0" instead of "sh")
That would actually do, wouldn't it? I mean a Nullhomotopy is not necesserally a straight-line homotopy. So maybe it's a loss of generality?

 

[sparkster]

What if your space is X=S^2? Wouldn't your homotopy travel outside X?

 

[morphism]

You're not guaranteed that "x+y" makes sense for x and y in X.