Abstract
For any flag simplicial complex $\Theta$ obtained by stellar subdividing the boundary of the cross polytope in edges, we define a flag simplicial complex $\Delta(\Theta)$ whose $f$-vector is the $\gamma$-vector of $\Theta$. This proves that the $\gamma$-vector of any such simplicial complex is the face vector of a flag simplicial complex, partially solving a conjecture by Nevo and Petersen. As a corollary we obtain that such simplicial complexes satisfy the Frankl-Füredi-Kalai inequalities.
Highlights
This paper relates to the theory of face enumeration of simplicial complexes
It gives a partial solution to a conjecture by Nevo and Petersen [13] on flag homology spheres, which are a particular class of simplicial complexes whose definition can be found in [4]
The conjecture is proven for a subclass of flag homology spheres, namely those that can be obtained by subdividing the boundary of the cross polytope in edges
Summary
It gives a partial solution to a conjecture by Nevo and Petersen [13] on flag homology spheres, which are a particular class of simplicial complexes whose definition can be found in [4]. The dual simplicial complex of a d-dimensional cube is the boundary complex of the cross polytope and is denoted Σd−1. Shaving a codimension two face of a simple polytope is equivalent to the stellar subdivision in an edge of the dual simplicial complex. For dimensions 4 and 5 the required complex is a graph with γ1(Θ) vertices and γ2(Θ) edges without triangles This construction coincides with the simplicial complexes defined in [1] when Θ is the dual simplicial complex of a flag nestohedron, a proof of which can be found in [2]. This theory has been developed and found applications in the framework of Toric topology, a new direction of algebraic topology; see [7]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
