Saturday November 16, 2013, at UCLA

 		A Day of Triangulations

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.

10:30 in Math Sciences 6620 (the Math Common Room): Coffee and greetings.

All talks are in Math Sciences 6627.

11:00: Rob Kirby, UC Berkeley. The 1968 UCLA torus trick epiphany and PL
triangulations of manifolds.

12-1:30 Lunch. (On your own. Plenty of options on campus and in Westwood.)

1:30: Bob Edwards, UCLA. Non-PL triangulations of manifolds exist.

2:45: Ron Stern, UCI (UCLA PhD 1973). Simplicial triangulations of high
dimensional manifolds and the homology cobordism group of homology 
3-spheres.

3:45: Refreshments in Math Sciences 6620.

4:15: Ciprian Manolescu, UCLA. Resolving the Triangulation Question in
high dimensions: The non-existence of certain homology 3-spheres.

5:15: Speakers' Panel: Further Discussion, Reminiscences and Open 
Problems.

6:00: Buffet Dinner in Math Sciences 6620.


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 Conjecture
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 
which 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.

Web site: http://www.math.ucla.edu/topology/tc13.html

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