Authors: Jojić, Duško
Nekrasov, Ilya
Panina, Gaiane
Živaljević, Rade 
Title: Alexander r-tuples and bier complexes
Journal: Publications de l'Institut Mathematique
Volume: 104
Issue: 118
First page: 1
Last page: 22
Issue Date: 1-Jan-2018
Rank: M24
ISSN: 0350-1302
DOI: 10.2298/PIM1818001J
We introduce and study Alexander r-tuples K = 〈K i 〉 i=1r of simplicial complexes, as a common generalization of pairs of Alexander dual complexes (Alexander 2-tuples) and r-unavoidable complexes of [3] and [11]. In the same vein, the Bier complexes, defined as the deleted joins K Δ* of Alexander r-tuples, include both standard Bier spheres and optimal multiple chessboard complexes (Section 2.2) as interesting, special cases. Our main results are Theorem 4.1 saying that (1) the r-fold deleted join of Alexander r-tuple is a pure complex homotopy equivalent to a wedge of spheres, and (2) the r-fold deleted join of a collective unavoidable r-tuple is (n - r - 1)-connected, and a classification theorem (Theorem 5.1 and Corollary 5.1) for Alexander r-tuples and Bier complexes.
Keywords: Alexander duality | Bier spheres | Chessboard complexes | Discrete Morse theory | Unavoidable complexes
Publisher: Mathematical Institute of the SASA
Project: Russian Science Foundation, grant 16-11-10039
Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 
Topology, geometry and global analysis on manifolds and discrete structures 

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