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Transformation of Curves on Two-Sided Surfaces

In: Papers on Group Theory and Topology

Author

Listed:
  • Max Dehn

Abstract

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 Poincaré (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.

Suggested Citation

  • Max Dehn, 1987. "Transformation of Curves on Two-Sided Surfaces," Springer Books, in: Papers on Group Theory and Topology, pages 183-199, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-4668-8_10
    DOI: 10.1007/978-1-4612-4668-8_10
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