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What is the fundamental group of the Möbius strip?
Is it given by $\{-1,1\}$ as the lemma of Synge supposes, or am I wrong and it does not apply there?

3 Answers 3

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The moebius strip is homotopy-equivalent to the circle, so has the same fundamental group which is $\mathbb Z$.

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    Thank you! That helped a lot!2012-10-27
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It is $\mathbb{Z}$. You can prove it via seeing the Möbius strip as a quotient of a square , with sides identified properly. Draw a diagonal dividing this square, and show that the Möbius strip deformation retracts onto this circle .

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    Instead of the diagonal, you could use the line through the center of the square and parallel to the unidentified edges.2013-05-24
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The fundamental group of the moebius strip is $\{a,b|a^2=b^2\}$. Cf. http://www2.math.ou.edu/~forester/5863S14/fsol.pdf

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    That is actually the fundamental group of the Klein bottle, although it can be constructed using Mobius bands.2017-12-09