Definition:Homotopy Group

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

Let $* : \map {\pi_n} {X, x_0} \times \map {\pi_n} {X, x_0} \to \map {\pi_n} {X, x_0}$ denote the concatenation of homotopy classes of paths.

That is, if $\overline {c_1}, \overline {c_2}$ are two elements of $\map {\pi_n} {X, x_0}$, then:

$\overline {c_1} * \overline {c_2} = \overline {c_1 \cdot c_2}$

where $\cdot$ denotes the usual concatenation of paths.

Then $\struct {\map {\pi_n} {X, x_0}, *}$ 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 $\map {\pi_1} {X, x_0}$ does not depend on $x_0$ and we just write $\map {\pi_1} X$.

Also see