Manolescu refutes the Triangulation Conjecture

This past week, Ciprian Manolescu posted a preprint on ArXiv proving (allegedly- I haven’t read the paper beyond the introduction) that the Triangulation Conjecture is false.

-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture.

This is big news. I feel it’s the last nail in the coffin of the Hauptvermutung. I’d like to tell you a little bit about the conjecture, and about Manolescu’s strategy, and what it has to do with low dimensional topology.

ICERM Fall 2013: Topology, geometry, and dynamics

I’ve mentioned before that the fall semester program at ICERM for 2013 will focus on computation in low-dimensional topology, geometry, and dynamics.   You can now apply to be a long-term visitor for this as a graduate student, postdoc, or other.   The deadline for the postdoctoral positions is January 14, 2013; the early deadline for everyone else is December 1, 2012 and the second deadline March 15, 2013.

There will also be three week-long workshops associated with this, so mark your calendars for these exciting events:

1. Exotic Geometric Structures. September 15-20, 2013.
2. Topology, Geometry, and Group Theory: Informed by Experiment. October 21-25, 2013.
3. Geometric Structures in Low-Dimensional Dynamics. November 18-22, 2013.

L-spaces and left-orderability of 3-manifold groups

Steve Boyer, Cameron Gordon, and Liam Watson have an interesting new preprint out today on the arXiv. In it, they posit:

Conjecture. An irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable.

The motivation here is as follows: An L-space is something whose Heegaard Floer homology is as simple as possible; such 3-manifolds have no taut foliations. A nice type of taut foliation are those that are R-covered, and in this case, the fundamental group of the 3-manifold inherits a left-order from the action of the leaf space. (I’m always assuming here that foliations are co-orientable.)

Of course, it’s not known whether every non-L-space has a taut foliation, and there are certainly non-R-covered foliations, so a reasonable initial reaction is that this conjecture isn’t very plausible. However, their paper outlines a surprising amount of evidence for it, and in this post I’ll give some more data in that direction.