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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Universal cover of Salvetti’s complex and topology of simplicial arrangements of hyperplanes
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by Luis Paris PDF
Trans. Amer. Math. Soc. 340 (1993), 149-178 Request permission

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 \[ M(\mathcal {A}) = {V_\mathbb {C}} - \left ({\bigcup \limits _{H \in \mathcal {A}} {{H_\mathbb {C}}} } \right ),\] 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.
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • 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