Lower bound theorems for simplicial and cubical complexes
Isabella Novik, Steven H. Klee · 2010
We study the classes of simplicial and cubical complexes. The common theme in studying these two classes of complexes is to discover lower bounds on their face numbers as a function of their underlying geometric structures. In the first part, we study the family of balanced simplicial complexes. We begin by bounding the number of faces in a balanced pseudomanifold in terms of the size of a minimal generating set of its fundamental group. We go on to study other classes of balanced simplicial complexes, namely the classes of doubly Cohen-Macaulay and Buchsbaum* complexes, which include balanced triangulations of spheres and orientable manifolds, respectively. We prove an analogue of the Barnette's Lower Bound Theorem (LBT) for these complexes by constructing a family of balanced spheres that simultaneously minimize all face numbers as a function of the dimension of the complex and its number of vertices. In the second part; we study the family of cubical pseudomanifolds. We begin by proving that the boundary complex of a d-dimensional cube has the minimal face numbers among all (d – 1)-dimensional cubical pseudomanifolds. We go on to study the class of cubical polytopes more generally, proving some special cases of the LBT for cubical complexes. We conclude by proving a version of the Dehn-Sommerville equations for cubical manifolds with boundary.