f-Vectors of Triangulated Balls

We describe two methods for showing that a vector cannot be the f-vector of a homology d-ball. As a consequence, we disprove a conjectured characterization of the f-vectors of balls of dimension five and higher due to Billera and Lee. We also provide a construction of triangulated balls with various f-vectors. We show that this construction obtains all possible f-vectors of three- and four-dimensional balls and we conjecture that this result also extends to dimension five.