Taut foliations from left orders, in Heegaard genus two

Abstract:

Suppose that \(M\) is a closed, connected, oriented three-manifold which is not graph. All previously known constructions of taut foliations on such \(M\) used branched surfaces. These branched surfaces come from sutured manifold hierarchies, following Gabai, come from spanning surfaces of knot exteriors, following Roberts, or come from one-vertex triangulations with foliar orientations, following Dunfield.

In this talk, we give a new construction that does not use branched surfaces. Instead, we build a taut foliation from the data of a Heegaard diagram for \(M\) and a left order on the fundamental group \(\pi_1(M)\). We glue an \(\mathbb{R}\)-transverse foliation (over a thickened Heegaard surface) to a pair of handlebody foliations; we then suitably cancel any singularities. For Heegaard diagrams satisfying mild conditions, this can be done reliably in Heegaard genus two. In some cases this construction can be extended to higher Heegaard genus. This helps explain numerical results of Dunfield: (i) tens of thousands of Heegaard-genus two hyperbolic L-spaces certifiably fail to admit fundamental group left orders and (ii) no hyperbolic L-space is known to admit a fundamental group left order.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

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Geometry and topology online

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The talks start five minutes after the hour. Talks are typically 55 minutes long, including time for questions.

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