Ph.D. (2011) Cornell University
Abstract: A fundamental invariant of a subdivision of a space into cells is its collection of face numbers or f-vector. A major area of study is understanding the possible f-vectors of various types of subdivisions. Most of this thesis works on characterizing the f-vectors of subdivisions of balls. We study two different types of subdivisions: simplicial complexes that are triangulations of balls and simplicial posets whose order complexes are balls.
For simplicial complexes 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 triangulated 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 triangulated three- and four-balls and we conjecture that this result also extends to dimension five.
For simplicial posets we use Stanley's idea of the face ring to develop a series of new conditions on the f-vectors of simplicial poset balls. We also present new methods for constructing simplicial poset balls with specific f-vectors. Combining this work with a result of Murai we give a complete characterization of the f-vectors of simplicial poset balls in even dimensions, as well as odd dimensions less than or equal to five.
The last chapter looks at surgeries of triangulated manifolds. First we derive formulas for the face numbers of a particular triangulation of the product of two simplicial complexes. Using these formulas, we determine how some surgeries change the face numbers of a manifold. This leads to new results about the f-vectors of certain manifolds.