Combinatorics Seminar

When: Sunday, December 2, 10am Where: Schreiber 309 Speaker: Eran Nevo, Ben Gurion University Title: On the generalized lower bound conjecture

Abstract:

The study of face numbers of polytopes is a classical problem. For a simplicial d-polytope P, its face numbers are conveniently encoded by the so called h-numbers h_0(P),...,h_d(P). In 1971, McMullen and Walkup posed the following conjecture, which is called "the generalized lower bound conjecture":
(1) If P is a simplicial d-polytope then h_0(P)<= h_1(P)<=...<= h_[d/2](P).
(2) Moreover, if h_{r-1}(P)=h_r(P) for some r <=d/2, then P can be triangulated without introducing simplices of dimension <=d-r.

Part (1) was proved by Stanley in 1980 using the hard Lefschetz theorem for projective toric varieties. Here we prove part (2), then generalize it to triangulated spheres admitting the weak Lefschetz property. The proof of part (2) uses algebraic, geometric and topological arguments. Time permitting, the picture for triangulated manifolds will be discussed as well.

Based on joint works with Satoshi Murai.