Surfaces that become isotopic after Dehn filling

TL;DR

This paper proves that generic Dehn fillings on a 3-manifold preserve the isotopy classes of essential surfaces, with only finitely many exceptions where surfaces can become isotopic.

Contribution

It establishes that essential surfaces remain distinct after generic Dehn fillings and characterizes the limited cases where isotopy classes can merge.

Findings

Essential surfaces are preserved under generic Dehn fillings.

No two distinct essential surfaces become isotopic after generic filling.

Only finitely many non-generic fillings cause isotopy among essential surfaces.

Abstract

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.