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Towards Lower Bounds for Complexity of 3-Manifolds: a Program

Sergei Anisov

math.GTarXiv:math/0103169

Abstract

For a 3-dimensional manifold M3, its complexity c(M3), introduced by S.Matveev, is the minimal number of vertices of an almost simple spine of M3; in many cases it is equal to the minimal number of tetrahedra in a singular triangulation of M3. An approach to estimating c(M3) from below for total spaces of torus bundles over S1, based on the study of theta-curves in the fibers, is developed, and pseudominimal special spines for these manifolds are constructed, which we conjecture to be their minimal spines. We also show how to apply some of these ideas to other 3-manifolds.

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