Changes

{{Math-a}}

{{Math-a}}

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 [[group|groups]] known as the homotopy groups and homology groups.  The most important of these is the 1^st 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 [[group]]s known as the homotopy groups and homology groups.  The most important of these is the 1st homotopy group, known as the '''fundamental group'''.

The concept of [[homotopy]], on which the fundamental group is based, is a type of equivalence between paths.  Two paths are said to be 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 said to be 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 main theorem for computing the fundamental group of topological spaces is the Seifert-van Kampen theorem.  This theorem gives a means to describe the fundamental group of a space obtained by gluing two other spaces together along subsets of these two spaces.  The computation of the fundamental group of the figure eight, described above, is the most elementary application of the Seifert-van Kampen theorem.

The main theorem for computing the fundamental group of topological spaces is the Seifert-van Kampen theorem.  This theorem gives a means to describe the fundamental group of a space obtained by gluing two other spaces together along subsets of these two spaces.  The computation of the fundamental group of the figure eight, described above, is the most elementary application of the Seifert-van Kampen theorem.

The theory of [[covering space|covering spaces]] provides another powerful tool for the computation of fundamental groups.

The theory of [[covering space]]s provides another powerful tool for the computation of fundamental groups.

== Applications ==

== Applications ==