Definition:Homotopy Group

From ProofWiki
Jump to navigation Jump to search


Let $X$ be a topological space.

Let $x_0 \in X$.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $\mathbb S^n \subseteq \R^{n + 1}$ be the $n$-sphere.

Let $\mathbf a \in \mathbb S^n$.

Let $\map {\pi_n} {X, x_0}$ be the set of homotopy classes relative to $\mathbf a$ of continuous mappings $f: \mathbb S^n \to X$ such that $\map f {\mathbf a} = x_0$.

Then $\map {\pi_n} {X, x_0}$ is the $n$th homotopy group of $X$.

The group operation of $\map {\pi_n} {X, x_0}$ is the concatenation operation.

Also defined as

Some sources define a homotopy group by instantiating the point $\mathbf a \in \mathbb S^n$ to be $\tuple {1, 0, \ldots, 0}$.

Also see

  • Results about homotopy groups can be found here.