Alexander $r$-tuples and Bier complexes

Abstract

Alexander $r$-tuples are introduced as a common generalization of pairs of Alexander dual complexes (Alexander $2$-tuples) and $r$-unavoidable complexes of Blagojević, 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.

arXiv commentThe authors acknowledge the support and hospitality of the Mathematisches Forschungsinstitut Oberwolfach, where this paper was initiated as a `Research in Pairs' project. In this version the presentation is improved and the list of references updated