fundamental group
Let $(X,{x}_{0})$ be a pointed topological space^{} (that is, a topological space^{} with a chosen basepoint ${x}_{0}$). Denote by $[({S}^{1},1),(X,{x}_{0})]$ the set of homotopy classes of maps $\sigma :{S}^{1}\to X$ such that $\sigma (1)={x}_{0}$. Here, $1$ denotes the basepoint $(1,0)\in {S}^{1}$. Define a product^{} $[({S}^{1},1),(X,{x}_{0})]\times [({S}^{1},1),(X,{x}_{0})]\to [({S}^{1},1),(X,{x}_{0})]$ by $[\sigma ][\tau ]=[\sigma \tau ]$, where $\sigma \tau $ means “travel along $\sigma $ and then $\tau $”. This gives $[({S}^{1},1),(X,{x}_{0})]$ a group structure and we define the fundamental group^{} of $(X,{x}_{0})$ to be ${\pi}_{1}(X,{x}_{0})=[({S}^{1},1),(X,{x}_{0})]$.
In general, the fundamental group of a topological space depends upon the choice of basepoint. However, basepoints in the same pathcomponent of the space will give isomorphic groups^{}. In particular, this means that the fundamental group of a (nonempty) pathconnected space is welldefined, up to isomorphism^{}, without the need to specify a basepoint.
Here are some examples of fundamental groups of familiar spaces:

•
${\pi}_{1}({\mathbb{R}}^{n})\cong \{0\}$ for each $n\in \mathbb{N}$.

•
${\pi}_{1}({S}^{1})\cong \mathbb{Z}$.

•
${\pi}_{1}(T)\cong \mathbb{Z}\oplus \mathbb{Z}$, where $T$ is the torus.
It can be shown that ${\pi}_{1}$ is a functor^{} from the category of pointed topological spaces to the category of groups. In particular, the fundamental group is a topological invariant, in the sense that if $(X,{x}_{0})$ is homeomorphic^{} to $(Y,{y}_{0})$ via a basepointpreserving map, then ${\pi}_{1}(X,{x}_{0})$ is isomorphic to ${\pi}_{1}(Y,{y}_{0})$.
It can also be shown that two homotopically equivalent pathconnected spaces have isomorphic fundamental groups.
Homotopy groups^{} generalize the concept of the fundamental group to higher dimensions^{}. The fundamental group is the first homotopy group, which is why the notation ${\pi}_{1}$ is used.
Title  fundamental group 
Canonical name  FundamentalGroup 
Date of creation  20130322 11:58:44 
Last modified on  20130322 11:58:44 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  16 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 57M05 
Classification  msc 55Q05 
Classification  msc 20F34 
Synonym  first homotopy group 
Related topic  Group 
Related topic  Curve 
Related topic  EtaleFundamentalGroup 