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'fundamental group'
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| Title of object: |
fundamental group |
| Canonical Name: |
FundamentalGroup |
| Type: |
Definition |
| Created on: |
2001-11-14 16:00:46 |
| Modified on: |
2006-10-07 11:06:57 |
| Classification: |
msc:55Q05, msc:20F34, msc:57M05 |
| Synonyms: |
fundamental group=first homotopy group |
Revision comment (for changes between this and next version):
Preamble:
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Content:
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 $\sigma\colon S^{1} \to X$
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 \emph{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 path-component of the space
will give isomorphic groups.
In particular, this means that the fundamental group of a (non-empty) path-connected space is well-defined, up to isomorphism,
without the need to specify a basepoint.
Two homotopically equivalent path-connected spaces have the same fundamental group.
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 basepoint preserving map,
then $\pi_1(X,x_0)$ is isomorphic to $\pi_1(Y,y_{0})$.
Examples of the fundamental groups of some familiar spaces are: $\pi_{1}(\mathbb{R}^n)\cong\{0\}$ for each $n$,
$\pi_{1}(S^1)\cong\mathbb{Z}$
and $\pi_{1}(T)\cong \mathbb{Z}\oplus\mathbb{Z}$, where $T$ is the torus.
The concept of the fundamental group can be generalized to higher dimensions, giving the series of homotopy groups.
The fundamental group is the first homotopy group. |
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