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On transformations of special spines
by
Artem U. Macovetsky
Chelyabinsk State University, Russia
There are different methods for presenting 3-manifolds. These are, for examples, triangulations of manifolds and Heegaard decompositions of manifolds.
It is known that for two triangulations of the same manifold there is a common star subdivision. Also, there is stable equivalence of any two Heegaard decomposition of same manifold.
Besides two above-mentioned methods for presenting 3-manifolds there is a well-known method for presenting 3-manifolds by special spines. The relation between 3-manifolds and spines was studed in the [C]. In [M] it was investigated T +/- 10, T +/- 12 moves for spines. We put next question. Is there result for special spines like two above-mentioned results for triangulations and Heegaard decompositions? The following theorem is answer.
Definition. A special spine Q of manifold M3 dominates a spine S of M3 if one can pass from S to Q by moves T0 and T2 .
Theorem 1. Let P and S be special spines of a manifold M3. Then there is a special spine Q of M3 such that Q dominates P and S.
Next theorem desribes a useful representation of special spines.
Theorem 2. Let P be special spine of a
manifold M3. Then there is a
special spine S of M3 such that:
1) boundary curve of every 2-component in the spine S is
simple closed curve;
2) one can pass from spine P to spine S by T2 moves only.
REFERENCES
[C] B.G.Casler, An imbedding for connected 3-manifolds with boundary, Proc. Amer. Math. Soc. 16(1965), 559-566.
[M] S.V. Matveev, Transformations of special spines and the Zeeman conjecture, Math. USSR Izvestia , Vol. 31, 2, 1988, 423-434.
Date received: May 29, 1999
Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacy-18.