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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'''.

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'''.

The fundamental group is based on the concept of [[homotopy]], a type of equivalence between paths.  Two paths are homotopic if one can be continuously deformed/moved/stretched into the other.  This stretching can't pass over a hole, so a characterization of homotopic paths yields a characterization of holes.

The concept of [[homotopy]], on which the fundamental group is based, is a type of equivalence between paths.  Two paths are homotopic if one can be continuously deformed/moved/stretched into the other.  This stretching can't pass over a hole, so a characterization of homotopic paths yields a characterization of holes.

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.

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.