Reducing and toroidal Dehn fillings on $3$-manifolds bounded by two tori

We show that if $M$ is a simple $3$-manifold bounded by two tori such that $M(r_1)$ is reducible and $M(r_2)$ is toroidal, then $\Delta(r_1,r_2)\le 2$, answering a question raised by Gordon. To do this, we first prove that there exists only one simple $3$-manifold having two Dehn fillings of distance $3$ apart one of which yields a reducible manifold and the other yields a $3$-manifold containing a Klein bottle.

Full Text (PDF format)

Published 1 January 2006