Abstract:
A crucial issue in lattice gravity is to control the number of triangulations of manifolds, at least asymptotically. Typically, one counts the number of different combinatorial types with respect to the number N of facets. For example, in two dimensions, there are exponentially many surfaces with bounded genus. Unfortunately, little is known about higher dimensions. How many 3-spheres are there? Which parameters should we restrict in order to achieve exponential bounds on 3-, or 4-manifolds? We will show how Forman's discrete Morse theory provides a new perspective for these problems.