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