ISSN:
1572-9192
Keywords:
cohomology
;
characteristic polynomial
;
Coxeter subspace arrangement
;
homotopy
;
homology
;
lexicographic shellability
;
signed graph
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Let $${\mathcal{D}}_{n,k} $$ be the family of linear subspaces of ℝn given by all equations of the form $$\varepsilon _1 x_{i_1 } = \varepsilon _2 x_{i_2 } = \cdot \cdot \cdot \varepsilon _k x_{i_k } ,$$ for 1 ≤ 〈 • • • 〈 i k ≤ i and $$\left( {\varepsilon _1 ,...,\varepsilon _k } \right)\varepsilon \left\{ { + 1, - 1} \right\}^k $$ Also let $${\mathcal{B}}_{n,k,h} $$ be $${\mathcal{D}}_{n,k} $$ enlarged by the subspaces $$x_{j_1 } = x_{j_2 } = \cdot \cdot \cdot x_{j_h } = 0,$$ for 1 ≤. The special cases $${\mathcal{B}}_{n,2,1} $$ and $${\mathcal{D}}_{n,2} $$ are well known as the reflection hyperplane arrangements corresponding to the Coxeter groups of type B nand D n respectively. In this paper we study combinatorial and topological properties of the intersection lattices of these subspace arrangements. Expressions for their Möbius functions and characteristic polynomials are derived. Lexicographic shellability is established in the case of $${\mathcal{B}}_{n,k,h,} 1 \leqslant h 〈 k$$ , which allows computation of the homology of its intersection lattice and the cohomology groups of the manifold $$\begin{gathered} {\mathcal{D}}_{n,2} \hfill \\ M_{n,k,h,} = {\mathbb{R}}^n \backslash \bigcup {{\mathcal{B}}_{n,k,h,} } \hfill \\ \end{gathered} $$ . For instance, it is shown that $$H^d \left( {M_{n,k,k - 1} } \right)$$ is torsion-free and is nonzero if and only if d = t(k − 2) for some $$t,0 \leqslant t \leqslant \left[ {{n \mathord{\left/ {\vphantom {n k}} \right. \kern-\nulldelimiterspace} k}} \right]$$ . Torsion-free cohomology follows also for the complement in ℂnof the complexification $${\mathcal{B}}_{n,k,h}^C ,1 \leqslant h 〈 k$$ .
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1022492431260
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