How does one go about computing the element of the fundamental group of a Möbius strip represented by the loop $(\cos 10\pi t, \sin 10\pi t)$.


HINT: You know that $\pi_1(M)\cong \mathbb{Z}$ via the isomorphism which takes $1\in\mathbb{Z}$ to the loop which goes around the Möbius strip once. Find how to express your loop as a product of the once-around loop and just pull back.

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  • $\begingroup$ I'm a little confused. My loop presumably loops around five times. So would the element of the fundamental group be $\mathbb{Z}^5$? Or would it be expressed in terms of words? $\endgroup$ – whistler123 Dec 8 '11 at 19:44
  • $\begingroup$ Now I'm confused a little to. $\mathbb{Z}^5$? Aren't you asking for the element of $\mathbb{Z}$ which corresponds to your loop? $\endgroup$ – Alex Youcis Dec 8 '11 at 19:46
  • $\begingroup$ Yeah. My loop goes around the circle five times, right? So the integer that corresponds to this loop is five? (I apologize if I'm entirely off the mark. This has been hard for me to understand) $\endgroup$ – whistler123 Dec 8 '11 at 19:57
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    $\begingroup$ Ok, so have that there is an isomorphism $f:\pi_1(M)\to \mathbb{Z}$ with the once-around loop mapping to $1$. So, if your loop is the $5$-fold product of the once-around loop, what does it map to in $\mathbb{Z}$? $\endgroup$ – Alex Youcis Dec 8 '11 at 20:13
  • $\begingroup$ Ah, it also maps to 1. $\endgroup$ – whistler123 Dec 8 '11 at 20:19

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