Rational homotopy type of subspace arrangements with a geometric lattice

Abstract

Let A = { x 1 , … , x n } \mathcal {A} = \{x_1, \dotsc , x_n\} be a subspace arrangement with a geometric lattice such that codim ⁡ ( x ) ≥ 2 \operatorname {codim}(x) \geq 2 for every x ∈ A x \in \mathcal {A} . Using rational homotopy theory, we prove that the complement M ( A ) M(\mathcal {A}) is rationally elliptic if and only if the sum x 1 ⊥ + … + x n ⊥ x_1^\perp + \dotso + x_n^\perp is a direct sum. The homotopy type of M ( A ) 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 ( A ) M(\mathcal {A}) is rationally elliptic, and show that most arrangements have a hyperbolic complement.