# Computing element of fundamental group of Möbius strip

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.
• 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? – whistler123 Dec 8 '11 at 19:44
• 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? – Alex Youcis Dec 8 '11 at 19:46
• 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 $\mathbb{Z}$? – Alex Youcis Dec 8 '11 at 20:13