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 Mosc. Math. J., 2006, Volume 6, Number 1, Pages 5–41 (Mi mmj233)

Lower bounds for transversal complexity of torus bundles over the circle

S. S. Anisov

Utrecht University

Abstract: For a 3-dimensional manifold $M^3$, its complexity $c(M^3)$, introduced by S. Matveev, is the minimal number of vertices of an almost simple spine of $M^3$; in many cases it is equal to the minimal number of tetrahedra in a singular triangulation of $M^3$. Usually it is straightforward to give an upper bound for $c(M)$, but obtaining lower bounds remains very difficult. We consider manifolds fibered by tori over the circle, introduce transversal complexity $tc(M)$ for such manifolds, and give a lower bound for $tc(M)$ in terms of the monodromy of the fiber bundle; this estimate involves a very geometric study of the modular group action on the Farey tesselation of hyperbolic plane. As a byproduct, we construct pseudominimal spines of the manifolds fibered by tori over $S^1$. Finally, we discuss some potential applications of these ideas to other 3-manifolds.

Key words and phrases: Complexity of 3-manifolds, $T^2$-bundles over $S^1$, Farey tesselation

DOI: https://doi.org/10.17323/1609-4514-2006-6-1-5-41

Full text: http://www.ams.org/.../abst6-1-2006.html
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Bibliographic databases:

MSC: Primary 57M99; Secondary 57M20, 57M50, 57R05, 57R15, 57R22
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Citation: S. S. Anisov, “Lower bounds for transversal complexity of torus bundles over the circle”, Mosc. Math. J., 6:1 (2006), 5–41

Citation in format AMSBIB
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