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Universal cover of Salvetti's complex and topology of simplicial arrangements of hyperplanes


Author: Luis Paris
Journal: Trans. Amer. Math. Soc. 340 (1993), 149-178
MSC: Primary 52B30; Secondary 32S25
DOI: https://doi.org/10.1090/S0002-9947-1993-1148044-X
MathSciNet review: 1148044
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Abstract: Let $ V$ be a real vector space. An arrangement of hyperplanes in $ V$ is a finite set $ \mathcal{A}$ of hyperplanes through the origin. A chamber of $ \mathcal{A}$ is a connected component of $ V - ({ \cup _{H \in \mathcal{A}}}H)$. The arrangement $ \mathcal{A}$ is called simplicial if $ { \cap _{H \in \mathcal{A}}}H = \{ 0\} $ and every chamber of $ \mathcal{A}$ is a simplicial cone. For an arrangement $ \mathcal{A}$ of hyperplanes in $ V$, we set

$\displaystyle M(\mathcal{A}) = {V_\mathbb{C}} - \left({\bigcup\limits_{H \in \mathcal{A}} {{H_\mathbb{C}}} } \right),$

where $ {V_\mathbb{C}} = \mathbb{C} \otimes V$ is the complexification of $ V$, and, for $ H \in \mathcal{A}$ , $ {H_\mathbb{C}}$ is the complex hyperplane of $ {V_\mathbb{C}}$ spanned by $ H$.

Let $ \mathcal{A}$ be an arrangement of hyperplanes of $ V$. Salvetti constructed a simplicial complex $ \operatorname{Sal}(\mathcal{A})$ and proved that $ \operatorname{Sal}(\mathcal{A})$ has the same homotopy type as $ M(\mathcal{A})$. In this paper we give a new short proof of this fact. Afterwards, we define a new simplicial complex $ \hat{\operatorname{Sal}}(\mathcal{A})$ and prove that there is a natural map $ p:\hat {\operatorname{Sal}}(\mathcal{A}) \to \operatorname{Sal}(\mathcal{A})$ which is the universal cover of $ \operatorname{Sal}(\mathcal{A})$. At the end, we use $ \hat{\operatorname{Sal}}(\mathcal{A})$ to give a new proof of Deligne's result: "if $ \mathcal{A}$ is a simplicial arrangement of hyperplanes, then $ M(\mathcal{A})$ is a $ K(\pi ,1)$ space." Namely, we prove that $ \hat{\operatorname{Sal}}(\mathcal{A})$ is contractible if $ \mathcal{A}$ is a simplicial arrangement.


References [Enhancements On Off] (What's this?)

  • [Br] K. S. Brown, Buildings, Springer-Verlag, New York, 1989. MR 969123 (90e:20001)
  • [Co] R. Cordovil, On the homotopy of the Salvetti complexes determined by simplicial arrangements, preprint.
  • [De] P. Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273-302. MR 0422673 (54:10659)
  • [FR] M. Falk and R. Randell, On the homotopy theory of arrangements, Complex Analytic Singularities, Advanced Studies in Pure Math., 8, North-Holland, Amsterdam, 1987, pp. 101-124. MR 894288 (88f:32045)
  • [JT] M. Jambu and H. Terao, Free arrangements of hyperplanes and supersolvable lattices, Adv. Math. 52 (1984), 248-258. MR 744859 (86c:32004)
  • [Lj] E. S. Ljapin, Semigroups, Transl. Math. Monographs, vol. 3, Amer. Math. Soc., Providence, R.I., 1974. MR 0352302 (50:4789)
  • [LW] A. T. Lundell and S. Weingram, The topology of $ CW$ complexes, Van Nostrand Reinhold, New York, 1969. MR 0238319 (38:6595)
  • [Or] P. Orlik, Introduction to arrangements, CBMS Regional Conf. Ser. in Math., no. 72, Amer. Math. Soc., Providence, R.I., 1989. MR 1006880 (90i:32018)
  • [Pa1] L. Paris, The covers of a complexified real arrangement of hyperplanes and their fundamental groups, preprint. MR 1243871 (94i:52014)
  • [Pa2] -, The Deligne complex of a real arrangement of hyperplanes, preprint.
  • [Pa3] -, Arrangements of hyperplanes with property $ D$, preprint.
  • [Sa1] M. Salvetti, Topology of the complement of real hyperplanes in $ {\mathbb{C}^N}$, Invent. Math. 88 (1987), 603-618. MR 884802 (88k:32038)
  • [Sa2] -, On the homotopy theory of complexes associated to metrical-hemisphere complexes, preprint.
  • [Te1] H. Terao, Arrangements of hyperplanes and their freeness. I, II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 293-320. MR 586451 (84i:32016a)
  • [Te2] -, Modular elements of lattices and topological fibration, Adv. Math. 62 (1986), 135-154. MR 865835 (88b:32032)
  • [We] A. Weil, Sur les théorèmes de De Rham, Comment. Math. Helv. 26 (1952), 119-145. MR 0050280 (14:307b)

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DOI: https://doi.org/10.1090/S0002-9947-1993-1148044-X
Article copyright: © Copyright 1993 American Mathematical Society

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