This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.Dieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden.Jojic, DuskoNekrasov, IlyaPanina, GaianeZivaljevic, Rade2019-06-2820161864-7596https://doi.org/10.34657/1949https://oa.tib.eu/renate/handle/123456789/3144We introduce and study Alexander r-tuples K = Kiir i=1 of simplicial complexes, as a common generalization of pairs of Alexander dual complexes (Alexander 2-tuples) and r-unavoidable complexes of [BFZ-1]. In the same vein, the Bier complexes, defined as the deleted joins K delta 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 4.3 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 collective unavoidable r-tuple is (n - r - 1)-connected, and a classification theorem (Theorem 5.1 and Corollary 5.2) for Alexander r-tuples and Bier complexes.application/pdfeng510Bier spheresAlexander dualitychessboard complexesunavoidable complexesdiscrete Morse theoryReport