A Day of Triangulations
Saturday, November 16, 2013
The day's program will be 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 will highlight some key aspects of these developments which are linked with UCLA.
All talks and coffee breaks will take place in the Math Sciences Building, in room 5200. The dinner will be in the Graduate Lounge (room 6620). For directions, see the interactive campus map.
This is an informal event. There is no financial support available for travel or accomodation, but there is no registration fee either. For lodging, we recommend the following hotels in the area: Tiverton House, UCLA Guest House, Royal Palace, Claremont Hotel, Hilgard House.
Here is some parking information. The closest lots to the math department are 8 and 9, and you can purchase a daily pass from the kiosk at the end of Westwood Boulevard.
Please contact Bob Brown (firstname.lastname@example.org) if you are planning to attend, or even if you haven't yet decided, so that we can be sure to order enough food.
10:15-11:00: Coffee and greetings.
11:00-12:00: Rob Kirby (UC Berkeley): The 1968 UCLA torus trick epiphany and PL triangulations of manifolds. Video
12:00-1:30: Lunch break - here is a map of places to eat in Westwood.
1:30-2:30: Bob Edwards (UCLA): Non-PL triangulations of manifolds exist. Video
2:45-3:45: Ron Stern (UC Irvine; UCLA Ph.D. 1973): Simplicial triangulations in high dimensions and the homology cobordism group of homology 3-spheres. Video
4:15-5:15: Ciprian Manolescu (UCLA): Resolving the Triangulation Question in high dimensions: The non-existence of certain homology 3-spheres. Video
5:15-6:00: Speakers' Panel: Further Discussion, Reminiscences and Open Problems.
6:00: Buffet dinner.
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.