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Bier spheres of extremal volume and generalized permutohedra

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arxiv 2108.00618 v1 pith:HZ634AGN submitted 2021-08-02 math.CO math.MG

Filip D. Jevtić , Rade T. Živaljević This is my paper · ORCID
classification math.COmath.MG
keywords bierspheresvolumecirccomplexgeneralizednaturalpermutohedra
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abstract

A Bier sphere $Bier(K) = K\ast_\Delta K^\circ$, defined as the deleted join of a simplicial complex and its Alexander dual $K^\circ$, is a purely combinatorial object (abstract simplicial complex). Here we study a hidden geometry of Bier spheres by describing their natural geometric realizations, compute their volume, describe an effective criterion for their polytopality, and associate to $K$ a natural fan $Fan(K)$, related to the Braid fan. Along the way we establish a connection of Bier spheres of maximal volume with recent generalizations of the classical Van Kampen-Flores theorem and clarify the role of Bier spheres in the theory of generalized permutohedra.

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  1. Simplicial vs. cubical spheres, polyhedral products and the Nevo-Petersen conjecture

    math.CO 2024-11 conditional novelty 7.0 of 10

    Flag Murai spheres are exactly the nerve complexes of four families of flag nestohedra, which implies the Nevo-Petersen conjecture for them.

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