Alexander $r$-tuples and Bier complexes

Duško Jojić, Ilya Nekrasov, Gaiane Panina, Rade Živaljević

We introduce and study Alexander $r$-tuples $\mathcal{K}=\langle K_i\rangle_{i=1}^r$ 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 $\mathcal{K}^*_\Delta$ 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~4.1 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.