Show that $[u]$ is a generator of $\pi(S^1,1)$

Question

Let $u:(I,\mathring I)\to(S^1,1)$ be the closed path $t\mapsto\exp(2\pi i t)$. Show that $[u]$ is a generator of $\pi(S^1,1)$.

I can see intuitively why this is true.

But I do not see how to prove it. Can someone give me hints?

What previous theorems should I use?

You can assume that I am familiar with all the previous theorems to this exercises. Which are all before page 54 from this book www.ugr.es/~acegarra/Rotman.pdf

(my apologies for not being clear)

Thank you

Answer 1

In Theorem 3.16 Rotman states that the degree function $\deg : \pi_1(S^1) \to \mathbb Z$ is an isomorphism. The degree of your closed path $u$ is $1$ because the inclusion $i : [0,1] \to \mathbb R$ is the unique lifting of $u$ with $i(0) = 0$.

Hence $[u]$ is mapped to the generator $1$ of $\mathbb Z$ so that $[u]$ is a generator of $\pi_1(S^1)$.