Category:Definitions/Fundamental Groups

This category contains definitions related to Fundamental Groups.
Related results can be found in Category:Fundamental Groups.

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.

This definition needs to be completed. In particular: For $\map {\pi_1} X$, a separate page Definition:Fundamental Group for Path-Connected Space You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding or completing the definition. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{DefinitionWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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 $\map {\pi_1} X$.