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, S² x S¹ and the nonorientable S² bundle over S¹. 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.