Isotopic Deformations of Geodesic Complexes on the 2-Sphere and on the Plane

Abstract

Two simplicial complexes, Ko and K1, are called isomorphic if their respective sets of vertices can be so numbered, Pi and Qi (i = 1, 2, ), that Qi, ... Qim is a cell of K1 when and only when Pi. * * * Pi,,, is a cell of Ko. We will then say that the vertices are similarly numbered. A complex on a euclidean 2-sphere will be referred to as geodesic if each of its 1-cells is an arc of great circle shorter than a semi-circle and each of its 2-cells is a spherical triangle less in area than a hemisphere. Every complex throughout this paper will be finite and will be non-singular, in the sense that no two different cells of any one complex will have a point in common. DEFORMATION THEOREM. Let Ko and K1 be a pair of isomorphic geodesic complexes on a euclidean 2-sphere, S. Let Pi and Qi (i = 1, * , n) be the verticeo, similarly numbered, of Ko and Kt respectively. If and only if the isomorphism3 between Ko and K1 can be extended into an orientation-preserving self-homeomorphism of S, it is possible to define, for every t (0 < t < 1), a geodesic complex, K1, with vertices Pi(t) (i = 1, * * *, n) in such a way that (1) Pi(0) = Pi and Pi(1) = Qi (i = 1, * * *, n) (2) K1 and Ko are isomorphic with vertices similarly numbered (3) as t increases from 0 to 1, Pi(t) traces a continuous curve on S (i = 1, * , n). The conclusion of this theorem asserts the existence of an isotopic deformation of Ko into K1 induced by a motion of the vertices, all cells remaining geodesic and non-degenerate throughout the motion. The proof of the theorem will occupy most of the remaining sections of the paper. Before commencing the proof, we state a criterion for the fulfilment of the necessary and sufficient condition involved in the theorem.