Applications of Klee's Dehn-Sommerville relations

TL;DR

This paper applies Klee's Dehn-Sommerville relations to establish new bounds on face numbers and Betti numbers of homology manifolds, confirming several conjectures and extending classical relations to manifolds with boundary.

Contribution

It proves Kalai's conjecture on lower bounds, verifies K"uhnel's conjecture on upper bounds, and extends Dehn-Sommerville relations to manifolds with boundary.

Findings

Proved Kalai's conjecture on f-vector lower bounds.

Verified K"uhnel's conjecture on Betti number upper bounds.

Extended Dehn-Sommerville relations to manifolds with boundary.

Abstract

We use Klee's Dehn-Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai's conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify K\"uhnel's conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove K\"uhnel's conjecture providing upper bounds on other Betti numbers of odd- and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee's Dehn-Sommerville relations and strengthen Kalai's result on the number of their edges.