Definition:Homotopy Group
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Definition
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
- Fundamental Group is Independent of Base Point for Path-Connected Space
- Higher Homotopy Groups are Abelian
- Results about homotopy groups can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): homotopy group
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): homotopy group