Definition:Homotopy Group
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Definition
Let $X$ be a topological space, and $x_0 \in X$.
Let $\mathbb S^n \subseteq \R^{n+1}$ be the $n$-sphere, and $a \in \mathbb S^n$.
Let $\pi_n \left({X, x_0}\right)$ be the set of homotopy classes relative to $a$ of continuous mappings $c: \mathbb S^n \to X$ such that $c(a) = x_0$.
Let $* : \pi_n \left({X, x_0}\right) \times \pi_n \left({X, x_0}\right) \to \pi_n \left({X, x_0}\right)$ denote the concatenation of homotopy classes of paths.
That is, if $\overline{c_1}, \overline{c_2}$ are two elements of $\pi_n \left({X, x_0}\right)$, then:
- $\overline{c_1} * \overline{c_2} = \overline{c_1 \cdot c_2}$
where $\cdot$ denotes the usual concatenation of paths.
Then $\left({\pi_n \left({X, x_0}\right), *}\right)$ is the $n$th fundamental group of $X$.
The first homotopy group is usually called the fundamental group when higher homotopy groups are not in sight.
For a path-connected manifold, by Fundamental Group is Independent of Base Point for Path-Connected Space, the isomorphism class of $\pi_1 \left({X, x_0}\right)$ does not depend on $x_0$ and we just write $\pi_1 \left({X}\right)$.