November Triangulation Conference
On Saturday, November 16, the UCLA mathematics department hosted a conference named A Day of Triangulations. Videos of the talks are now available on the conference website.
The program was devoted to the history and solution of the century-old Triangulation of Manifolds Question: Is an arbitrary topological manifold triangulable, i.e. homeomorphic to a simplicial complex?
The talks served to highlight some key aspects of these developments that are linked with UCLA. The speakers presenting were Rob Kirby, Bob Edwards, Ron Stern, and Ciprian Manolescu. For more information, see the conference web site at http://www.math.ucla.edu/~topology/tc13.html
Some Background and History: The first successful efforts on the Triangulation-of-Manifolds Question occurred in the 1920s when the 2-dimensional case was solved. Dimension 3 took another 30 years, finally being solved in the early 1950s. For the next 15 years the question remained open in dimensions >= 4, attracting ever more attention.
In 1968 a breakthrough happened at UCLA when Rob Kirby discovered the torus trick. He then used it to crack open the higher dimensional cases for so-called combinatorial, i.e. PL triangulations, working with L. Siebenmann.
Following the Kirby-Siebenmann work attention turned to the non-PL side of the Triangulation Question. The existence of non-PL triangulations was known to reduce to the Double Suspension Question for homology spheres: Is the 2-fold suspension of a homology sphere homeomorphic to a (genuine) sphere? This was solved affirmatively by Bob Edwards in 1974-76 for almost all cases. This provided non-PL triangulations for many manifolds that the Kirby-Siebenmann work had shown were not PL-triangulable. Jim Cannon completed the affirmative answer to the DSQ in 1977.
Still there remained many manifolds for which the triangulation question remained open. In the mid-1970s a broad theory of triangulations was developed by Ron Stern with coauthor Dave Galewski, and independently by Takao Matumoto. This reduced the high-dimensional triangulation problem to a question of the existence of a special class of homology 3-spheres.
In 1985 the first non-simplicially-triangulable manifolds were found in dimension four by Andrew Casson, following work of Mike Freedman. Finally in 2013 Ciprian Manolescu answered the Galewski-Stern-Matumoto question in the negative, thus showing the existence of non-triangulable manifolds in all dimensions five or higher.