Diagrams over Groups

Abstract

It was mentioned in the introduction that, in 1933, van Kampen [112] made a simple but extremely interesting observation about the universal possibility of interpreting geometrically the deduction of consequences of relations in groups (which, however, remained unnoticed until the mid-sixties). It revealed a new connection between the ideas of combinatorial topology and combinatorial group theory. It was really new, and must not be confused with, for example, the well-known representation of an arbitrary group as the fundamental group of a 2-dimensional topological space. In order to emphasize this distinction, we turn attention firstly to the fact that van Kampen’s lemma deals exclusively with planar complexes (the usefulness of other surfaces was discovered later), and not with arbitrary 2-complexes. Secondly, van Kampen’s lemma combines ideas not only from abstract algebra and topology, but also from mathematical logic, since van Kampen diagrams adequately reflect the process of deducing consequences of given relations in groups.