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The Homology of the Real Complement of a $k$-parabolic Subspace Arrangement

Abstract : The $k$-parabolic subspace arrangement, introduced by Barcelo, Severs and White, is a generalization of the well known $k$-equal arrangements of type-$A$ and type-$B$. In this paper we use the discrete Morse theory of Forman to study the homology of the complements of $k$-parabolic subspace arrangements. In doing so, we recover some known results of Björner et al. and provide a combinatorial interpretation of the Betti numbers for any $k$-parabolic subspace arrangement. The paper provides results for any $k$-parabolic subspace arrangement, however we also include an extended example of our methods applied to the $k$-equal arrangements of type-$A$ and type-$B$. In these cases, we obtain new formulas for the Betti numbers.
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Christopher Severs, Jacob A. White. The Homology of the Real Complement of a $k$-parabolic Subspace Arrangement. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.437-448. ⟨hal-01186311⟩

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