Rational homotopy type of subspace arrangements with a geometric lattice

Details Rational homotopy type of subspace arrangements with a geometric lattice Rational homotopy type of subspace arrangements with a geometric lattice

2007-05-10

arxiv.org/abs/0705.1449v1

Mathematics

Algebraic Topology

7 pages, to be published in Proceedings of the AMS

Scientific paper

Let A be a subspace arrangement with a geometric lattice such that codim(x) > 1 for every x in A. Using rational homotopy theory, we prove that the complement M(A) is rationally elliptic if and only if the sum of the orthogonal subspaces is a direct sum. The homotopy type of M(A) is also given: it is a product of odd dimensional spheres. Finally, some other equivalent conditions are given, such as Poincare duality. Those results give a complete description of arrangements (with geometric lattice and with the codimension condition on the subspaces) such that M(A) is rationally elliptic, and show that most arrangements have an hyperbolic complement.

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