Fundamental group of the circle

Remarks

The proof uses the covering map $p \colon \R \to S^1$, $p(t) = e^{2\pi i t}$. By the path lifting property, a loop $\gamma$ at $1$ lifts to a unique path $\tilde\gamma$ in $\R$ with $\tilde\gamma(0) = 0$, and $\tilde\gamma(1)$ is an integer, the degree of $\gamma$. Homotopy lifting shows that path homotopic loops have the same degree, so the inverse map $[\gamma] \mapsto \tilde\gamma(1)$ is well defined. The lift of $\omega_n$ is $s \mapsto ns$. The lift of a concatenation is obtained by translating and joining lifts, which gives the homomorphism property. The degree is the topological form of the winding number. Consequences include the Brouwer fixed point theorem in dimension $2$ and the fundamental theorem of algebra.

Going deeper

For a student

This is the first nontrivial computation of a fundamental group, and the model for computations through covering spaces. It shows that the circle is not contractible and that it is not a retract of the closed disk. With the Seifert–van Kampen theorem it gives the fundamental groups of wedges of circles, of surfaces and of many CW complexes. It is also the case $n = 1$ of the computation $\pi_n(S^n) \cong \Z$.