fundamental group
Let be a pointed topological space (that is, a topological space
with a chosen basepoint ).
Denote by
the set of homotopy classes of maps
such that .
Here, denotes the basepoint .
Define a product
by ,
where means “travel along and then ”.
This gives a group structure
and we define the fundamental group
of
to be .
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.
Here are some examples of fundamental groups of familiar spaces:
-
•
for each .
-
•
.
-
•
, where is the torus.
It can be shown that 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 is homeomorphic
to via a basepoint-preserving map,
then is isomorphic to .
It can also be shown that two homotopically equivalent path-connected 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 is used.
Title | fundamental group |
Canonical name | FundamentalGroup |
Date of creation | 2013-03-22 11:58:44 |
Last modified on | 2013-03-22 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 |