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Rational homotopy type of subspace arrangements with a geometric lattice

Author: Gery Debongnie
Journal: Proc. Amer. Math. Soc. 136 (2008), 2245-2252
MSC (2000): Primary 55P62
Published electronically: February 14, 2008
MathSciNet review: 2383531
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Abstract: Let $ \mathcal{A} = \{x_1, \dotsc, x_n\}$ be a subspace arrangement with a geometric lattice such that $ \codim(x) \geq 2$ for every $ x \in\mathcal{A}$. Using rational homotopy theory, we prove that the complement $ M(\mathcal{A})$ is rationally elliptic if and only if the sum $ x_1^\perp + \dotso + x_n^\perp$ is a direct sum. The homotopy type of $ M(\mathcal{A})$ is also given: it is a product of odd-dimensional spheres. Finally, some other equivalent conditions are given, such as Poincaré duality. Those results give a complete description of arrangements (with a geometric lattice and with the codimension condition on the subspaces) such that $ M(\mathcal{A})$ is rationally elliptic, and show that most arrangements have a hyperbolic complement.

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Additional Information

Gery Debongnie
Affiliation: Université Catholique de Louvain, Departement de Mathematique, Chemin du Cyclotron, 2, B-1348 Louvain-la-Neuve, Belgium

Received by editor(s): January 31, 2007
Published electronically: February 14, 2008
Additional Notes: The author is an “Aspirant” of the “Fonds National pour la Recherche Scientifique” (FNRS), Belgium.
Communicated by: Paul Goerss
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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