Homotopy groups of spheres

Definition

Let $n \ge 1$ , and let $S^n = \set{x \in \R^{n+1} : \norm{x} = 1}$ be the unit $n$ -sphere with base point $s_0 = (1, 0, \ldots, 0)$ . For $k \ge 1$ , the group $\pi_k(S^n)$ is the homotopy group $\pi_k(S^n, s_0)$ . The homotopy groups of spheres are the groups $\pi_k(S^n)$ for all $k, n \ge 1$ .

Remarks

Changing the base point gives isomorphic groups, because $S^n$ is path-connected; this is why it is usually omitted. Example: $\pi_1(S^1) \cong \Z$ , and the class of a loop is its winding number. Example: $\pi_k(S^n) = 0$ for $1 \le k < n$ . Give $S^n$ the CW structure with one $0$ -cell and one $n$ -cell. By cellular approximation, a based map $S^k \to S^n$ is homotopic to a map into the $k$ -skeleton, which is a point.

Non-example: the groups are not the homology groups of spheres. The homology $H_3(S^2)$ is zero, but $\pi_3(S^2) \cong \Z$ , generated by the Hopf map.

Further values: $\pi_n(S^n) \cong \Z$ for all $n \ge 1$ , $\pi_k(S^1) = 0$ for $k \ge 2$ , and $\pi_{n+1}(S^n) \cong \Z/2$ for $n \ge 3$ .

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Homotopy group concept
Homotopy groups of spheres definition

Tag	`212MDDPN8A`
URI	`m2://212MDDPN8A`
Kind	definition
Writing	W0 stub
MSC	55Q40, 55Q05
Defines	Homotopy groups of spheres
Uses	Homotopy group
Changed	2 November 2026

Homotopy groups of spheres

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here