Alexander r-Tuples and Bier Complexes

Abstract

We introduce and study Alexander $r$-Tuples $\mathcal{K} = \langle K_i \rangle ^r_{i=1}$ 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 $\mathcal{K}^*_\Delta$ 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 4.3 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.