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# Mathematical Research Letters

## Volume 13 (2006)

### Number 2

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

Pages: 287 – 306

DOI: http://dx.doi.org/10.4310/MRL.2006.v13.n2.a9

#### Author

#### Abstract

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.