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.