Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 48, No. 1, pp. 27-34 (2007)

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Topological Criteria for \boldmath$k$-formal Arrangements

Stefan O. Toh\v aneanu

Department of Mathematics, Texas A&M University, College Station, TX 77843, e-mail:

Abstract: We prove a criterion for $k-$formality of arrangements, using a complex constructed from vector spaces introduced in [BT]. As an application, we give a simple description of $k$-formality of graphic arrangements: Let $G$ be a connected graph with no loops or multiple edges. Let $\Delta$ be the flag (clique) complex of $G$ and let $H_{\bullet}(\Delta)$ be the homology of the chain complex of $\Delta$. If $\mathcal A_G$ is the graphic arrangement associated to $G$, we will show that $\mathcal A_G$ is $k-$formal if and only if $H_i(\Delta)=0$ for every $i=1,\ldots,k-1$. \item[BT] K. A. Brandt; H. Terao: {\em Free Arrangements and Relation Spaces}. Discrete Comput.Geom. \textbf{12} (1994), 49--63.

Keywords: hyperplane arrangement, graph, chain complex, flag complex

Classification (MSC2000): 52C35; 18G35

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Electronic version published on: 14 May 2007. This page was last modified: 27 Jan 2010.

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