 
Summary: FLAG SUBDIVISIONS AND VECTORS
CHRISTOS A. ATHANASIADIS
Abstract. The vector is an important enumerative invariant of a flag
simplicial homology sphere. It has been conjectured by Gal that this
vector is nonnegative for every such sphere and by Reiner, Postnikov
and Williams that it increases when is replaced by any flag simplicial
homology sphere which geometrically subdivides . Using the nonneg
ativity of the vector in dimension 3, proved by Davis and Okun, as
well as Stanley's theory of simplicial subdivisions and local hvectors,
the latter conjecture is confirmed in this paper in dimensions 3 and 4.
1. Introduction
This paper is concerned with the face enumeration of an important class of
simplicial complexes, that of flag homology spheres, and their subdivisions.
The face vector of a homology sphere (more generally, of an Eulerian simpli
cial complex) can be conveniently encoded by its vector [10], denoted
(). Part of our motivation comes from the following two conjectures (we
refer to Section 2 for all relevant definitions). The first, proposed by Gal
[10, Conjecture 2.1.7], can be thought of as a Generalized Lower Bound
Conjecture for flag homology spheres (it strengthens an earlier conjecture
by Charney and Davis [8]). The second, proposed by Postnikov, Reiner and
