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Re: A homotopically trivial path
Posted:
Oct 10, 2006 7:26 AM
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From: mskirvin@gmail.com
Newsgroups: sci.math
Subject: Re: A homotopically trivial path
William Elliot wrote:
> > Let p:I -> S be a loop at a, ie a path with p(0) = a = p(1).
> > Let p_a be the constant path at a, ie p(I) = {a}. I = [0,1].
> > Assume p is homotopic to p_a.
> > Is p homotopic to p_a with respect to, relative to { 0,1 }?
> It's not necessarily the case that your homotopy is rel {0, 1}. For
> example, consider loops in the circle S^1. If f and g are any two
> loops at 1 (= 1 + 0i, regarding S^1 as the unit circle in the
> complexes), then they are homotopic, as can be seen by the homotopy
> F:I^2 -> S^1 given by F(s, t) = f((1-t)s)*g(ts). So, F(s, 0) = f(s)
> and F(s, 1) = g(s), showing f and g are homotopic.
Astounding.
> In particular, any loop in S^1 is homotopic to the constant loop at
> 1, but since the fundamental group of S^1 is nontrivial, not every
> loop is homotopic to the constant loop rel {0, 1}.
Now however if S were contractible, in which case it would contract to a,
tho not necessarily strongly contract to a, could the conclusion be made?
> More generally, it seems like the above homotopy can be used to show
> that any two loops centered at the identity in a topological monoid
> are homotopy equivalent. I'm guessing there are other topological
> monoids out there with nontrivial fundamental group that could
> furnish further counterexamples.
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