Max Dehn

Lectures on Group Theory

We begin our considerations with the generation and geometric representation of the group Σ3! of permutations of three things. We have already shown that this group may be generated by a transposition s1, which exchanges the first and second terms, and a cyclic permutation s2 of all three terms, which replaces the first element by the second, the second by the third, and then the third by the first. When we apply the transposition s1 twice, or the cycle s2 three times, then in each case we come back to the identity, hence

Transformation of Curves on Two-Sided Surfaces

\text{s}_\text{1}^\text{2} = 1\text{ and s}_\text{2}^\text{3} = 1.

Über unendliche diskontinuierliche Gruppen

In combinatorial topology, topological concepts are represented by arithmetic concepts. Thus, in principle, all problems of combinatorial topology are reduced to arithmetic problems. However, this reduction is of no use for the resolution of the problems in most cases, because the corresponding arithmetic problems have little relation to known results or methods. This is especially true of all problems in which homotopic transformations are considered non-trivial or, as one can also say, in which the exceedingly numerous and hard to visualize constructions of simply connected polyhedra of different dimensions come into play. In an earlier work* I have attempted to represent this construction arithmetically in an understandable way for two-dimensional polyhedra. In doing so, I showed that homotopy problems yield arithmetic problems which fall outside the extensive domain of group theory. They concern more general operations which are difficult to investigate, the totality of which I have covered by the name “games”.

Über die Topologie des dreidimensionalen Raumes

The problem we shall deal with in what follows is one of the simplest of topology: given two closed curves on a closed two-sided surface, to decide whether one may be “transformed” into the other by a continuous deformation. The solution of this problem for surfaces of genus p > 1 by means of “polygon groups” and hence on the basis of the metric of the hyperbolic plane is evident, and is indicated e.g. by Poincare (Rend. Circ. Mat. Pal. 1904), also developed more precisely by me in Math. Ann. 71*. In the same work I have given a method for deciding the question purely topologically without the help of the metric. However, in the foundation of this method I have made essential use of properties of figures in the hyperbolic plane. For surfaces of genus p = 0 and p = 1 the solution of the problem is very simple: in the first case all curves are transformable into each other, in the second case the fundamental group is abelian, and each curve is transformable into one which traverses a fixed curve C m times and a fixed curve Γ μ times, and these numbers m and μ are independent of the particular transformation, so that the transformation problem is solved.

Papers on Group Theory and Topology

In the previous section we have discussed the generation of groups by given operations, and we have repeatedly come up against infinite groups thereby. In this chapter we shall now deal exclusively with such groups, and use them to make a series of important applications to analysis situs or topology.

Die beiden Kleeblattschlingen

In the present work, we topologically investigate closed curves (knots) in ordinary (i.e. hyperspherical) space (Chapter II), as well as three-dimensional space in general (Chapter III). Preceding these discussions are the necessary general considerations (Chapter I). In fact in §1 we consider groups of discrete operations, namely those constructed from a finite number of generating operations, among which a finite number of relations are given.