Two continuous maps $f, g: X \to Y$ are homotopic if there's a continuous $H: X \times [0, 1] \to Y$ with $H(\cdot, 0) = f$ and $H(\cdot, 1) = g$. Homotopy is an equivalence relation.

The fundamental group $\pi_1(X, x_0)$ has elements = homotopy classes of loops based at $x_0$, with concatenation as the group operation. Examples:

• $\pi_1(\mathbb R^n) = 0$ (everything contracts).
• $\pi_1(S^1) = \mathbb Z$ — count windings.
• $\pi_1(T^2) = \mathbb Z \oplus \mathbb Z$ — two independent generators.
• $\pi_1(S^n) = 0$ for $n \geq 2$.

Higher homotopy groups $\pi_n(X) = [S^n, X]$ are abelian. Computing them is generally hard — $\pi_3(S^2) = \mathbb Z$ (Hopf), $\pi_n(S^n) = \mathbb Z$. Homology $H_n$ is a more computable cousin: abelian invariants counting independent $n$-cycles modulo boundaries.

Interactive: loops on a torus

Three loops on a torus: contractible (red), $a$-cycle (gold), $b$-cycle (blue). The latter two generate $\pi_1(T^2) = \mathbb Z^2$ and are not homotopic to a point.