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The '''fundamental group''' is a basic construction of [[algebraic topology]]. For a given surface, such as a torus or even the (humorously named) dunce cap, the fundamental group is the set of loops in the space that have the following characteristic: each loop can be continuously deformed to one another, a relation known as [[homotopy]].

One of the most important problems of topology (and [[algebraic topology]] in particular) is the precise and rigorous characterization of the "holes" in a space.  This characterization is in terms of some [[groups]] known as the homotopy groups and homology groups. The most important of these is the 1st homotopy group, known as the '''fundamental group'''.

To a [[topological space]] <math>X</math>, the fundamental group construction associates an algebraic object which describes the set of "holes" in the space.

 

Given a [[topological space]] <math>X</math>, the fundamental group is an associated algebraic object which describes the paths that are not homotopic, and hence describes the set of "holes" in the space.

The fundamental group is useful to distinguish between different topological spaces: for example, to prove rigorously that a surface with two holes cannot be smoothly deformed into a surface with three holes, one can check that these two spaces have different fundamental groups and so must be distinct.  The fundamental group captures information about the set of holes in a space by looking at the set of loops drawn on the space.

The fundamental group is useful to distinguish between different topological spaces: for example, to prove rigorously that a surface with two holes cannot be smoothly deformed into a surface with three holes, one can check that these two spaces have different fundamental groups and so must be distinct.  The fundamental group captures information about the set of holes in a space by looking at the set of loops drawn on the space.