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.
|Date of creation||2013-03-22 11:58:44|
|Last modified on||2013-03-22 11:58:44|
|Last modified by||yark (2760)|
|Synonym||first homotopy group|