An Algorithm to Classify 3-Manifolds?

Resenhas IME-USP 1995, Vol. 2, No. 2, 209-217.

An Algorithm to Classify 3-Manifolds?

Sóstenes L. Lins

Department of Mathematics
Federal University of Pernambuco
Recife, PE
Brazil
E-mail address: sostenes@dmat.ufpe.br

Abstract

Combinatorial topology makes unlimited use of refinements. These refinements translate into an unlimited amount of data to describe objects like 3-manifolds. As a result, procedures to treat the homeomorphism problem, by combinatorial means, become unfeasible. In this article we describe a finite algorithm which attains a combinatorial classification of 3-gems (tridimensional manifolds encoded by graphs), see [S.L. Lins, Gems, Computers and Attractors for 3-Manifolds, Number 5 in Knots and Everything, World Scientific, 1995] or Chapter 13 of [L.H. Kauffman and S.L. Lins, Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds, Number 134 in Annals of Mathematical Studies, Princeton University Press, 1994]. Up to the level studied (3-gems of 30 vertices), the combinatorial classification coincides with the topological one. A general question arises as at what level the combinatorial classes no longer coincides with the classes of homeomorphisms. The hope that this coincidence can always hold enable us to enunciate a conjecture that would computationally classify closed 3-manifolds. The central point in our approach is that the type of refinements in 3-gems that we permit (the U-move) are only applied in conjunction with some horizontal moves (the TS-moves that do not increase the size of the objects) and by others (

-moves) which decrease its size in such a way that the final 3-gem has no more vertices than the original one.

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Last modified: Tue Oct 31 17:51:12 GMT 1995