Atlas: A Way for Coding 3-Manifolds by Means of Atoms by V. O. Manturov

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Low-Dimensional Topology and Combinatorial Group Theory
July 31 - August 7, 1999
Chelyabinsk State University
Chelyabinsk, Russia

Organizers
Sergei V. Matveev

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A Way for Coding 3-Manifolds by Means of Atoms
by
V. O. Manturov
Moscow State University

Consider an atom as a 2-manifold with 4-graph on it, dividing the manifold into cells that can be coloured properly with two colours. Each triangulation of a 3-manifolds generates so-called Casler's complex, i.e. 2-frame of cell complex, dual to the initial triangulation; if 2-cells of Casler complex can be coloured with three colours in a proper way, then

1-cells of the initial triangulation can be coloured with the same three colours properly,

Casler's complex can be considered as a triple of atoms in the following way: removing all 2-cells coloured with one colour, we get that the rest of the complex has to be an atom; (So, removing the first, the second and the third colours, we receive three atoms, that is the sens of the word "triple"). Each atom of the triple can be reconstructed by the any other one. So, if an atom is given , we can construct the triple of atoms and after that the proper coloured triangulation of some 3-manifold.

Main Theorem. Each 3-manifold has a properly coloured triangulation.

Finally we get:

Each coloured triangulation of 3-manifold generates a triple of atoms.

Manifold can be uniquely restored by such an atom.

Besides, I proved the the theorem, that each orientable manifold can be generated by an orientable atom and studied the question whether an atom generates some menifold. Here each ätom" Casler's complex is closely connected with a triple of atoms and can be reconstructed by each of them. Besides, I set some simple properties of 3-manifolds (orientability, fundamental group, etc) in the terms of atom theory.

References:

[Ma1] V.O.Manturov. Bifurcations, Atoms, and Knots (in Russian)// Vestnik of the MSU, ser. math., 1999, to be appeared

[Ma2] V.O.Manturov. Atoms, Vertical Atoms, Chord diagrams, and Knots. Enumeration of Atoms of Low Complexity Using Mathematica 3.0(in Russian)// Topological Methods in Hamiltonian Systems Theory, Factorial ed., Moscow., 1998 pp. 203-212

[Ma3] V.O.Manturov. On One Way for Coding 3-Manifolds.(in Russian)// Trudy Conf. of Young Scientists, MSU, 1999, to be appeared

[Cas65] B.G.Casler.An imbedding theorem for connected 3-manifolds with boundary// Proc. Amer. Math. Soc. Volume 16, 1965 pp. 559-566

[Fom] A.T.Fomenko. The Theory of Multidimensional Integrable Hamiltonian Systems (with Arbitrary many Degrees of Freedom). Molecular Table of All Integrable Systems with Two Degrees of Freedom// ADVANCES IN SOVIET MATHEMATICS Volume 6, 1991.

Date received: June 11, 1999