Homotopy Group

For $n\ge 1$, the $n$th Homotopy Group of a pointed topological space $(X,x)$ (written ${\pi }_n(X,x)$) is defined as the group with:

• objects being homotopy classes of continuous maps $(I^n,\partial I^n)\to (X,x)$, mapping the boudary $\partial I^n$ to the base-point $x$.
• multiplication being path concatenation along the first axis.

Properties

• $n\ge 2⟹{\pi }_n(X,x)\text{is Abelian}$.
• ${\pi }_n(S^n)\cong \mathbb{Z}$
• If $X$ is path connected then its homotopy group doesn’t depend on base-point.

See also

Fundamental Group - homotopy group when $n=1$

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Key result

$${\pi }_n(S^n)\cong \mathbb{Z}$$