Universidad Complutense de Madrid
E-Prints Complutense

Reduction of the Poincaré conjecture to other geometric conjectures

Impacto

Downloads

Downloads per month over past year



Montesinos Amilibia, José María (1972) Reduction of the Poincaré conjecture to other geometric conjectures. Revista Matemática Hispanoamericana, 32 . pp. 33-51. ISSN 0373-0999



Abstract

Throughout his paper, the author uses "orientable manifold'' to mean a compact connected orientable 3-manifold without boundary. Such a manifold is known to be a ramified covering over a link of the 3-sphere, in which the ramification index of each singular point is ≤2. If the covering has n leaves, suppose that there are m points of index 2 and 2m points of index 1; such a covering is of type (m,n−2m). The author's main theorem states: Every orientable manifold is a ramified covering of type (1,n−2).
He also uses the notion of a "link with a colouring of type (m,n−2m)''; these are intimately related to ramified coverings of type (m,n−2m). He conjectures that every link having a colouring of type (1,n−2) is "separable'', a term too complicated to define here. With this conjecture and his main theorem, he enunciates two further theorems and a second conjecture to show that his two conjectures, if true, would imply the Poincaré hypothesis for 3-manifolds. The author adds a note in proof to say that his first conjecture is false, as will be shown in a forthcoming paper by R. H. Fox. It therefore seems unnecessary to detail the conjectures in this review.


Item Type:Article
Subjects:Sciences > Mathematics > Topology
ID Code:21876
Deposited On:14 Jun 2013 17:55
Last Modified:12 Dec 2018 15:14

Origin of downloads

Repository Staff Only: item control page