A Reduction of the Poincare Conjecture to Group Theoretic Conjectures

Abstract

POINCARE CONJECTURE. Any simply-connected closed3 3-manifold is a 3-sphere, i.e., any homotopy 3-sphere is a 3-sphere. This paper presents a reduction of the above conjecture to the two conjectures below. Let N be an orientable closed3 surface of genus p ! 2, and let Ai, Bi, i = 1, * * *, p, be a fundamental system of N with base point o, (see No. 4)4. As it is well known, N has certain hyperbolic metrics, imposed on it by Poincare, and therefore there are geodesics on N, (see No. 3). Let now Xi and Yi be the primary simple closed geodesics on N, corresponding to Ai and Bi respectively, i = 1, * * *, p, (Lemma (12.2)). Then Xi and Yi have only one point qi in common (Nos. 3 and 12), any two of the pairs Xi, Yi different from each other, are disjoint, and if we cut N along the X and Y's, we obtain a 2-sphere with p holes having boundaries Xi YiXT1 y-1. Let X', i = 1, ... , p, be primary simple closed geodesics on N, any two of which different from each other are disjoint, and such that X' is homologous to Yi on N. Let oi be some common point of Xi and X'. Let now j be a fixed natural number > 1 and ? p. By an isotopic deformation of each one of the pairs Xi, Yi, i = 1, ... , p, on N, such that qj moves on XJ, we obtain a new fundamental system Aji, Bji, i = 1, ..., p, of N with base point oj, such that the carrier of Ajj is precisely Xi. Then