Lower bounds for transversal complexity of torus bundles over the circle
S. S. Anisov
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
MSC: Primary 57M99; Secondary 57M20, 57M50, 57R05, 57R15, 57R22
Received: December 30, 2005
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
\paper Lower bounds for transversal complexity of torus bundles over the circle
\jour Mosc. Math.~J.
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