Surveys in 3-Manifold Topology and Geometry

Presents a collection of surveys to bring subfields of 3-manifold topology up-to-date. These include Ricci flow-with-surgery on 3-manifolds stemming from the geometrization theorem; minimal surface techniques applied to the study of Heegaard splittings of 3-manifolds; and the theory of foliations and contact structures on sutured 3-manifolds.

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In the last half-century, tremendous progress has been made in the study of 3-dimensional topology. Many revolutions in 3-manifold topology during this period have come from outside of the field, including Kleinian groups, minimal surfaces, foliations, von Neumann algebras, gauge theory, mathematical physics, 4-manifolds, symplectic topology, contact topology, Riemannian geometry and PDEs, number theory, dynamics, and geometric group theory. The influx of ideas from neighboring fields has made the subject of 3-manifolds (and more generally low-dimensional topology) a very rich subject, creating subfields such as quantum topology. But this also means that there is a tremendous amount of background material for a novitiate in the subject to learn and master.

This volume is a collection of surveys meant to bring certain subfields of 3-manifold topology up-to-date. These include: Richard Bamler on Ricci flow-with-surgery on 3-manifolds stemming from Perelman's work on the geometrization theorem; Tobias Colding, David Gabai, and Daniel Ketover on minimal surface techniques applied to the study of Heegaard splittings of 3-manifolds, including the resolution of the Pitts–Rubinstein conjecture; Vincent Colin and Ko Honda on the theory of foliations and contact structures on sutured 3-manifolds; John Etnyre and Lenhard Ng on Legendrian contact homology of knots; Sang-Hyun Kim and Genevieve Walsh on hyperbolic groups with planar limit sets in relation to Kleinian groups; Marc Lackenby on algorithms in knot theory and 3-manifold topology, including results on computational complexity; Yi Liu and Hongbin Sun on the resolution of the virtual Haken conjecture, including subgroup separability, degree one maps between 3-manifolds, and torsion in the homology of covers; Mahan Mj on Cannon–Thurston maps following his resolution of Question 14 from Thurston's problem list; and Jean-Marc Schlenker on renormalized volume of Kleinian groups and its relation to other notions of volume.