Hello,(adsbygoogle = window.adsbygoogle || []).push({});

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:

[itex]

\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}

[/itex]

(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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Free homotopy

**Physics Forums | Science Articles, Homework Help, Discussion**