Combinatorial stratification of complex arrangements

Björner, Anders; Ziegler, Günter M.

doi:10.1090/S0894-0347-1992-1119198-9

Combinatorial stratification of complex arrangements
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by Anders Björner and Günter M. Ziegler

J. Amer. Math. Soc. 5 (1992), 105-149

DOI: https://doi.org/10.1090/S0894-0347-1992-1119198-9

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Abstract:

We present a method for discretizing complex hyperplane arrangements by encoding their topology into a finite partially ordered set of “sign vectors.” This is used in the following ways: (1) A general method is given for constructing regular cell complexes having the homotopy type of the complement of the arrangement. (2) For the case of complexified arrangements this specializes to the construction of Salvetti [S]. We study the combinatorial structure of complexified arrangements and the Salvetti complex in some detail. (3) This general method simultaneously produces cell decompositions of the singularity link. This link is shown to have the homotopy type of a wedge of spheres for arrangements in ${\mathbb {C}^d},\;d \geq 4$. (4) The homology of the link and the cohomology of the complement are computed in terms of explicit bases, which are matched by Alexander duality. This gives a new, more elementary, and more generally valid proof for the Brieskorn-Orlik-Solomon theorem and some related results. (5) Our setup leads to a more general notion of “$2$-pseudoarrangements,” which can be thought of as topologically deformed complex arrangements (retaining only the essential topological and combinatorial structure). We show that all of the above remains true in this generality, except for the sign patterns of the Orlik-Solomon relations.

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