The fundamental group of the Möbius strip

Question

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?

Answer 1

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

Answer 2

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 .