Abstract
In 2009, David Bachman introduced the notions of topologically minimal surfaces and topological index to generalize classes of surfaces such as strongly irreducible and incompressible surfaces. Topologically minimal surfaces have been useful in problems that deal with stabilization, amalgamation, and isotopy of Heegaard splittings of 3-manifolds and bridge spheres for knots. In this thesis, we will introduce the theory of topologically minimal surfaces and study Heegaard splittings from this perspective. In particular, we prove that Heegaard surfaces of genus g > 1 for the 3-sphere are topologically minimal, which disproves the conjecture that the 3-sphere contains no topologically minimal surfaces.
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