  We describe two methods for showing that a vector can not 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.
