Surfaces that become isotopic after Dehn filling

David Bachman, Ryan Derby-Talbot, Eric Sedgwick

Published 2010-01-24, updated 2013-02-27Version 2

We show that after generic filling along a torus boundary component of a 3-manifold, no two closed, 2-sided, essential surfaces become isotopic, and no closed, 2-sided, essential surface becomes inessential. That is, the set of essential surfaces (considered up to isotopy) survives unchanged in all suitably generic Dehn fillings. Furthermore, for all but finitely many non-generic fillings, we show that two essential surfaces can only become isotopic in a constrained way.