Speaker:  Eron Nevo
Title:  A characterization of simplicial polytopes with $\g_2=1$.
Time: 4:15 PM, Monday, April 7, 2008 (Note the earlier time.)
Place:  Malott 205

Abstract:
Let $v$ and $e$ be the numbers of vertices and edges, respectively, of a simplicial polytope $P$. Define $g_2(P)=e-dv+\binom{d+1}{2}$. Barnette proved that  $g_2(P)$ is nonnegative.  Kalai proved that the simplicial polytopes, or more generally homology spheres, with $\g_2=0$ are the stacked polytopes ($d\geq 4$). We characterize the $\g_2=1$ case.  It turns out they are the polytopes obtained by stacking over either a join of two simplices whose dimensions add up to $d$, or a join of a polygon with a $(d-2)$-simplex ($d\geq 4$).  The proof uses rigidity theory of graphs and Alexander duality.  Related problems will be mentioned.   All the needed notions  will be explained in the talk.

April 3, 2008