Ed Swartz,
Dpt. Mathematics - Cornell University Face enumeration on manifolds |

In the over 100 years since the discovery of the Euler-Poincaré formula there have been tremendous advances in the understanding of the geometry and topology of manifolds. However, the enumerative properties of triangulations remain largely mysterious. For instance, there are no manifolds in dimension five or higher whose f-vector is completely understood.

In 1970 Walkup
provided characterizations of all possible f-vectors of the three and
four-sphere, the three-dimensional projective plane, S^{2} x S^{1} and
the nonorientable S^{2} bundle over S^{1}. Our main goal will be to present
results and techniques which allow a description of the f-vectors of
several 4-dimensional manifolds. Along the way we will examine
implications of the g-conjecture for spheres on manifolds, and the
existence of 2-neighborly triangulations.