Vol. 259, No. 2, 2012

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Flag subdivisions and γ-vectors

Christos A. Athanasiadis

Vol. 259 (2012), No. 2, 257–278

The γ-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 Δ and by Reiner, Postnikov and Williams that it increases when Δ is replaced by any flag simplicial homology sphere that geometrically subdivides Δ. Using the nonnegativity of the γ-vector in dimension 3, proved by Davis and Okun, as well as Stanley’s theory of simplicial subdivisions and local h-vectors, the latter conjecture is confirmed in this paper in dimensions 3 and 4.

flag complex, homology sphere, simplicial subdivision, flag subdivision, face enumeration, γ-vector
Mathematical Subject Classification 2010
Primary: 05E45
Secondary: 05E99
Received: 11 January 2012
Revised: 28 May 2012
Accepted: 4 June 2012
Published: 3 October 2012
Christos A. Athanasiadis
Division of Algebra-Geometry
Department of Mathematics
National and Kapodistrian University of Athens
15784 Athens