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I'm looking at 29 pages. When he does the does the calculation of the fundmental group of circle.

Was wondering is there a easier way using covering spaces to prove this. As I'm trying to find a quick and easy way to do this. But, I can't seem to find one. I found an easy way to show that sphere and higher spheres are zero. But, that was using van kampen theorem and it won't work for the circle.

So was wondering what is a neat way to prove this.

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    $t \mapsto e^{2\pi i t}$ is a covering map $\mathbb{R} \to S^1$.2012-03-12
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    @t.b. I know, but don't see the point of covering spaces. Like I can't see how you use it to do any calculations. It's just heres a cover.2012-03-12
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    You can use covering spaces for all sorts of things. For example, Reidemeister torsions.2012-03-12
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    "I'm looking at 29 pages. When he does the does the calculation of the fundmental group of circle." is not the clearest way to start a question. You mean you're looking at page 29 of Hatcher?2012-03-12
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    @simplicity: as an example of their utility, when is the last time you have thought of angle measure as being anything other than a real number, possibly together with the notion that it should be modulo $2 \pi$?2012-04-29
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    Does simplicity mean to ask: is there an easier way **than** using covering spaces to prove this? If so the answer is clearly "yes" as given in my answer below, namely to use the van Kampen theorem for a set of base points, $2$ in the case of the circle.2012-05-27

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