Given a certain class of simple polyhedral complexes P and the associated Borel space BT P we compute the E2-term of the Unstable Adams Novikov Spectral Sequence for BT P through a range. As a result, through a range, the higher homotopy groups of BT P are isomorphic to the homotopy groups of a wedge of spheres whose dimensions depend on the combinatorics of P. This paper provides a unified approach to attacking the problem of computing the higher homotopy groups of complements of arbitrary complex coordinate subspace arrangements. We extend all higher homotopy group computations in the cases where the homotopy type of a complement of a complex coordinate subspace arrangement is unknown. If K is a simplicial complex that defines a triangulation of a sphere that is dual to a simple convex polytope P, then, in many cases, the homotopy groups of the quasi-toric manifold M2n(λ) can be computed through a range that was previously unknown. As an application, the homotopy type of a family of moment angle complexes ZK will be determined.
Homology, Homotopy and Applications, Vol. 10 (2008), No. 1, pp.437-479.