Definition:Fundamental Group/Definition 2

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Definition

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.



This definition needs to be completed.
In particular: For $\map {\pi_1} X$, a separate page Definition:Fundamental Group for Path-Connected Space
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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

  • 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 $1935$.


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