Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Description

In the early days of topology, manifolds and their invariants were often studied via triangulations. Since the manifolds themselves, and not their combinatorial structure, are the real objects of interest in topology, there was a growing desire to get away from triangulations, and therefore in the 1930's and 40's algebraic tools gradually replaced the combinatorial ones.

While triangulations always remained of interest to discrete geometers and geometric and $PL$ topologists, the emergence of computers has subtly changed the general situation. It is now possible (at least in principle) to study compact manifolds and compute their invariants on a machine.

In this talk, I will report on joint work with Anders Bjoerner, Ekkehard Koehler and Wolfgang Kuehnel on developing computer methods for experimentation with triangulations. In particular, we had in mind to explicitly construct minimal or otherwise optimal triangulations.

Using a heuristic, based on bistellar flips that locally modify the triangulation of a manifold, minimal triangulations were found for S^2xS^2, S^3xS^2, S^3xS^3 and RP^4. Also we obtained a 16-vertex triangulation of the Poincar'e homology 3-sphere, which is the starting point for a series of non-PL d-spheres with d+13 vertices for d >= 5.

Moreover, our heuristic is helpful to recognize the homeomorphism type of a manifold. For many examples of vertex-transitive combinatorial manifolds with few vertices, which we obtained by enumeration, this was carried out successfully.