Rational homotopy type of subspace arrangements with a geometric lattice
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- by Gery Debongnie PDF
- Proc. Amer. Math. Soc. 136 (2008), 2245-2252 Request permission
Abstract:
Let $\mathcal {A} = \{x_1, \dotsc , x_n\}$ be a subspace arrangement with a geometric lattice such that $\operatorname {codim}(x) \geq 2$ 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.References
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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
- Email: debongnie@math.ucl.ac.be
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2245-2252
- MSC (2000): Primary 55P62
- DOI: https://doi.org/10.1090/S0002-9939-08-09312-X
- MathSciNet review: 2383531