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Preprints 1995

On subspace arrangements of type $\mathcal D$
Citation key Report-489-1995
Author Eva Maria Feichtner and Dmitry N. Kozlov
Year 1995
Number 489
Institution Technische Universität Berlin, Institut für Mathematik
Abstract Let $\dnk$ denote the subspace arrangement formed by all linear subspaces in $R^n$ given by equations of the form $$\epsilon_1 x_i_1=\epsilon_2 x_i_2=\dots= \epsilon_k x_i_k\hspace1in$$ where $1łeq i_1<\dots <i_kłeq n$ and $(\epsilon_1,\dots, \epsilon_k) \in\+1,-1\^k$. Some important topological properties of such a subspace arrangement depend on the topology of its intersection lattice. In previous work on a larger class of subspace arrangements by Björner & Sagan \citeBS the topology of the intersection lattice $L(\dnk)=\pnkk$ turned out to be a particularly interesting and difficult case. We prove in this paper that $\pure(\pnkk)$ is shellable, hence that $\pnkk$ is shellable for $k>\fracn2$. Moreover we prove that $\widetilde H_i(\pnkk)=0$ unless \mbox$i\equiv n-2\,$ $(\textmod \, k-2)$ or $i\equiv n-3 \, (\textmod \, k-2)$, and that $\widetilde H_i(\pnkk)$ is free abelian for $i\equiv n-2\, (\textmod \, k-2)$. In the special case of $\pkkk$ we determine homology completely. Our tools are EC-shellability introduced in \citeKoz1 and a spectral sequence method for the computation of poset homology first used in \citeHan. We state implications of our results on the cohomology of the complement of the considered arrangements.
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