Homotopy Group

Let $(X, x_0)$ be a pointed topological space.

Definition. For $n \geq 1$, the $n$-th homotopy group $\pi_n(X, x_0)$ is the set of basepoint-preserving homotopy classes of maps $(S^n, *) \to (X, x_0)$, i.e.,

equipped with the group operation induced by the pinch map $S^n \to S^n \vee S^n$. Equivalently, $\pi_n(X, x_0) = [(I^n, \partial I^n),\, (X, x_0)]$: homotopy classes of maps from the $n$-cube sending its entire boundary to $x_0$. For $n = 0$, $\pi_0(X)$ is the set of path-components of $X$ (with no group structure in general). For $n = 1$, this recovers the fundamental group.

Proposition. For $n \geq 2$, $\pi_n(X, x_0)$ is abelian.

Proof sketch. The cube model $I^n$ with $n \geq 2$ admits two independent composition directions. The Eckmann–Hilton argument shows that these two operations on $\pi_n$ coincide and are commutative. $\square$

Proposition. A continuous map $f: (X, x_0) \to (Y, y_0)$ induces group homomorphisms $f_*: \pi_n(X, x_0) \to \pi_n(Y, y_0)$ for all $n \geq 1$, and this assignment is functorial.

Proposition. If $p: E \to B$ is a fibration with fiber $F = p^{-1}(b_0)$, there is a long exact sequence

Examples.

• $\pi_n(S^n) \cong \mathbb{Z}$ for all $n \geq 1$, detected by the degree of a map.
• $\pi_k(S^n) = 0$ for $k < n$: the sphere $S^n$ is $(n-1)$-connected.
• $\pi_3(S^2) \cong \mathbb{Z}$, generated by the Hopf fibration $S^3 \to S^2$.
• $\pi_n(\mathbb{R}^k, 0) = 0$ for all $n \geq 1$: Euclidean space is contractible.

Definition. $X$ is $n$-connected if $\pi_k(X, x_0) = 0$ for all $0 \leq k \leq n$. In particular, $0$-connected means path-connected, and $1$-connected means simply connected.