On Heegaard splittings of glued 3-manifolds

We introduce a new technique for finding lower bounds on the Heegaard genus of a 3-manifold obtained by gluing a pair of 3-manifolds together along an incompressible torus or annulus. We deduce a number of inequalities, including one which implies that $t(K_1# K_2)\geq \max {t(K_1),t(K_2)}$, where $t(-)$ denotes tunnel number, $K_1$ and $K_2$ are knots in $S^3$, and $K_1$ is $m$-small. This inequality is best possible. We also provide an interesting collection of examples, similar to a set of examples found by Schultens and Wiedmann, which show that Heegaard genus can stay persistently low under the kinds of gluings we study here.