"Lower bound theorems for simplicial and cubical complexes"

Steve Klee
University of Washington

Monday, September 14, 2009
4:00 pm, 845 Patterson Office Tower

Abstract:

It is well-known that the d-simplex has the minimal face numbers among all simplicial d-polytopes. Barnette's Lower Bound Theorem for simplicial polytopes, a much stronger result, establishes that a stacked d-polytope on n vertices has the minimal face numbers among all convex d-polytopes on n vertices. We will discuss two generalizations of these theorems. A balanced simplicial complex of dimension d-1 is one whose graph is d-colorable. We prove that a connected sum of d-dimensional cross polytopes has the minimal face numbers among all balanced (d-1)-spheres on n vertices. On the other hand, a cubical polytope is one whose faces are cubes, as opposed to simplices. We will give a precise combinatorial definition of cubical complexes and prove that any cubical decomposition of a d-dimensional pseudomanifold requires at least 2^{d+1} vertices. This generalizes a result of Blind and Blind.