Van Kampen diagram
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High Quality Content by WIKIPEDIA articles! In the mathematical area of geometric group theory, a van Kampen diagram is a planar diagram used to represent the fact that a particular word in the generators of a group given by a group presentation represents the identity element in that group. The notion of a van Kampen diagram was introduced by Egbert van Kampen in 1933. This paper appeared in the same issue of American Journal of Mathematics as another paper of van Kampen, where he proved what is now known as the Seifert?van Kampen theorem. The main result of the paper on van Kampen diagrams, ...
High Quality Content by WIKIPEDIA articles! In the mathematical area of geometric group theory, a van Kampen diagram is a planar diagram used to represent the fact that a particular word in the generators of a group given by a group presentation represents the identity element in that group. The notion of a van Kampen diagram was introduced by Egbert van Kampen in 1933. This paper appeared in the same issue of American Journal of Mathematics as another paper of van Kampen, where he proved what is now known as the Seifert?van Kampen theorem. The main result of the paper on van Kampen diagrams, now known as the van Kampen lemma can be deduced from the Seifert?van Kampen theorem by applying the latter to the presentation complex of a group. However, van Kampen did not notice it at the time and this fact was only made explicit much later. Van Kampen diagrams remained an underutilized tool in group theory for about thirty years, until the advent of the small cancellation theory in the1960s, where van Kampen diagrams play a central role. Currently van Kampen diagrams are a standard tool in geometric group theory.
