The following are some of the open problems/questions that have been discussed in blog posts:

2-Manifolds

Can the action of the mapping class group of a surface on the unit tangent bundle of the surface be realized by diffeomorphisms?  See Ian Agol’s comment on 7/3/08.

Which (infinite, non-cyclic) subgroups of the mapping class group of a surface can be realized by subgroups of the diffeomorphism or self-homeomorphism group?  See 7/3/08.

Does Hempel’s algorithm for constructing paths in the curve complex produce geodesics? See 1/16/08.

3-Manifolds

Given a Heegaard surface, characterize the set of loops in its curve complex that are homotopy trivial in the ambient manifold.  (Originally due to Yair Minsky) Characterize the loops that are isotopic to geodeiscs.  See 7/14/08.

Is the subgroup of a mapping class group generated by the mapping class groups of two handlebodies an amalgamated free product of the two groups?  (Originally due to Yair Minsky) See 7/14/08.

When is the non-minimal Heegaard surface produced by lifting a minimal bridge surface to the double branched cover reducible?  See 7/9/08.

Given the commensurator of (the exteriors of) two knots in S^3, how do the fillings obtained by the lifts of their meridians compare?  In particular, what happens for Aitchison and Rubinstein’s dodecahedral knots?  See 6/30/08.

Given a 3-manifold that can be embedded in R^4, can it be embedded so that the restriction of the height function is a Morse function inducing a minimal genus Heegaard splitting?  See 6/13/08.

Is every minimal layered triangulation of a lens space minimal among all triangulations? See 5/21/08.

Is there a connection between the existence of a tight/fillable/etc. contact structure and the existence of a high distance Heegaard splitting? See 5/20/08.

Does every 3-manifold have a Heegaard splitting such that the set of primitive disks has maximal dimension in the curve complex? See 4/29/08.

Prove something about Heegaard splittings using splitting homomorphisms. See 3/13/08.

Is there a hyperbolic 3-manifold with a Heegaard splitting such that each handlebody induces a redundant set of generators for the fundamental group? See 3/3/08.

If, given two non-isotopic Heegaard splittings of a fixed 3-manifold, we take the connect sum of each with a fixed Heegaard splitting of a second 3-manifold, can the resulting Heegaard splittings be isotopic? See 3/3/08.

Does the existence of a tight contact structure imply the existence of a strongly irreducible Heegaard splitting or vice versa? See 2/21/08.

Classify the set of all open book decompositions that induce Heegaard splittings in a fixed isotopy class. See 2/21/08.

Is every mapping class group of a 3-manifold isomorphic to a subgroup of a surface mapping class group? See 2/15/08.

Does the short exact sequence associated with the mapping class group of a Heegaard splitting always split? See 2/15/08.

Is there a hyperbolic 3-manifold whose rank is strictly less than its genus? (And if so, how big can the difference be?) See 2/11/08.

Does every cusped finite-volume hyperbolic three-manifold admit a geometric triangulation? (Here the ideal tetrahedra must have positive volume.) Suggested by Saul Schleimer.

Given an irreducible Heegaard splitting of a 3-manifold with a torus boundary such that the Heegaard splitting is stabilized for infinitely many Dehn fillings, is the Heegaard splitting necessarily PADed? See 2/5/08.

Is there a direct proof (i.e. not using geometrization) that 3-manifolds with Heegaard splittings of distance three or greater are hyperbolic? See 11/29/07.

Are the mapping class groups of Heegaard splittings homologically stable (under stabilization)? See 11/19/07.

Is there a 3-manifold with an irreducible, weakly reducible Heegaard splitting of non-minimal genus? See 11/19/07.

Is there an algorithm to determine the Heegaard genus of a 3-manifold? See 11/19/07.

Does every hyperbolic 3-manifold have a finite cover that is Haken/fibered/positive first Betti number? See 11/19/07.

Can the mapping class group of an irreducible Heegaard splitting act trivially on the fundamental groups of the handlebodies of the splitting? See 11/19/07.

Characterize 3-manifolds with trivial Casson invariant. See 11/19/07.

List all closed hyperbolic 3-manifolds with volume less than … See 11/18/07.

Knots and Links

Is it possible to level an unknotting tunnel with respect to a level unknotted torus while keeping the knot in minimal bridge position with respect to the torus? See 1/24/08.

Are there sets of vertices in the width complex with bounded width but infinite diameter? See 1/15/08.

Is the width complex (with all higher dimensional cells) homotopy equivalent to Hatcher’s space of knots? See 1/15/08.

If an unknotting tunnel in a hyperbolic knot is isotopic to a geodesic in the knot complement, does it define a unique pair of meridians for the knot relative to the tunnel, and if so are these the same as those defined by leveling the tunnel with respect to a bridge sphere? See 1/11/08.

Is every knot in S^3 with a non-trivial lens space surgery a Berge knot? See 11/19/07 and 1/19/08.

Is every unknotting tunnel for a hyperbolic tunnel number one knot isotopic to a geodesic? See 11/18/07.

Characterize all tunnel number one fibered knots. See 11/19/07.

Misc.

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