Using Straight-Line Homotopies to Construct Free Homotopies

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SUMMARY

A path-connected space X is simply-connected if and only if any two loops in X are free homotopic. The discussion centers on the use of straight-line homotopies to construct free homotopies, specifically through the example of defining a homotopy H_f(t,s) involving loops u and v and a connecting curve w. The inquiry raises concerns about the validity of using straight-line homotopies in all cases, particularly in relation to the potential for the homotopy to exit the space X, such as when X is the 2-sphere S^2. The conclusion emphasizes the need for caution when applying straight-line homotopies in certain topological contexts.

PREREQUISITES
  • Understanding of path-connected spaces and simply-connected spaces in topology.
  • Familiarity with the concept of free homotopy and its implications.
  • Knowledge of homotopy theory and the construction of homotopies.
  • Basic proficiency in LaTeX for mathematical expressions.
NEXT STEPS
  • Study the properties of simply-connected spaces in algebraic topology.
  • Explore the concept of nullhomotopies and their applications in homotopy theory.
  • Research the implications of homotopies in different topological spaces, particularly S^2.
  • Learn about alternative methods for constructing homotopies beyond straight-line approaches.
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Mathematicians, topologists, and students studying algebraic topology who are interested in the nuances of homotopy theory and its applications in various spaces.

littleHilbert
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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:<br /> \begin{equation} H_f(t,s):=<br /> \begin{cases} (1-3s)u(t)+3sa,\ \text{for $0\leqslant{}s\leqslant\frac{1}{3}$}; \\<br /> w(3s-1),\ \text{for $\frac{1}{3}\leqslant{}s\leqslant\frac{2}{3}$}; \\<br /> (3-3s)b+(3s-2)v(t),\ \text{for $\frac{2}{3}\leqslant{}s\leqslant1$}. <br /> \end{cases}<br /> \end{equation}<br />
(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?
 
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What if your space is X=S^2? Wouldn't your homotopy travel outside X?
 
You're not guaranteed that "x+y" makes sense for x and y in X.
 

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