  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 $M^3$. An approach to estimating $c(M^3)$ from below
for total spaces of torus bundles over $S^1$, 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.
