Abstract

A basic measure of the complexity of a manifold is the minimum number of simplices in a triangulation.  I will explain how using L^2 cohomology, hyperbolization and related ideas, one can get bounds on this for certain lens spaces (and homotopy lens spaces) – joint work with G. Lim.  Assuming a bound on local geometry this result could be deduced analytically from work of Calabi and Cheeger-Gromov, but the general case seems to require (for me) additional geometric input, essentially proving a result about positively curved manifolds by nonpositively curved geometry!  If there is time, I will try to put this into a conjectural perspective on how simplicial complexity interacts with surgery theory (and why the standard einsatz of simplicial complexity being a proxy for volume is wrong) and explain some obstacles (arising in work with F. Manin) to realizing such a vision.