Definition:Fundamental Group
Definition
Definition $1$
Let $\struct {X, x_0}$ be a pointed topological space with base point $x_0$.
The fundamental group $\map {\pi_1} {X, x_0}$ of $X$ at the base point $x_0$ is the set of homotopy classes of loops with base point $x_0$ with multiplication of homotopy classes of paths.
Definition $2$
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.
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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 known as
The fundamental group of a topological space $T$ is also known more explicitly as the fundamental homotopy group of $T$.
The fundamental group of $T$ is also known as the Poincaré group of $T$, for Henri Poincaré.
Also see
- Fundamental Group is Group
- Definition:Fundamental Group Functor
- Definition:Homotopy Group
- Fundamental Group is Independent of Base Point for Path-Connected Space
- Results about fundamental groups can be found here.
Historical Note
The concept of the fundamental group was defined by Henri Poincaré in $1895$.
The definition was extended to $n > 1$ by Eduard Čech in $1932$ and Witold Hurewicz in $1935$.