Alexander $r$-tuples and Bier complexes

TL;DR

This paper introduces Alexander r-tuples and Bier complexes, generalizing dual complexes and unavoidable complexes, and proves their topological properties including homotopy equivalence to wedges of spheres.

Contribution

It defines Alexander r-tuples and Bier complexes, providing a unified framework and classifying these complexes with new topological results.

Findings

The r-fold deleted join of Alexander r-tuples is homotopy equivalent to a wedge of spheres.

The r-fold deleted join of a collectively unavoidable r-tuple is highly connected.

Complete classification of Alexander r-tuples and Bier complexes is provided.

Abstract

Alexander $r$-tuples are introduced as a common generalization of pairs of Alexander dual complexes (Alexander $2$-tuples) and $r$-unavoidable complexes of Blagojevi\'{c}, Frick and Ziegler. The associated "Bier complexes" include both the Bier spheres and "optimal multiple chessboard complexes" as interesting, special cases. Our main result is Theorem 4.3 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 collectively unavoidable $r$-tuple is $(n-r-1)$-connected. We also give a complete classification (Theorem 5.1 and Corollary 5.2) of Alexander $r$-tuples and Bier complexes.