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# Homology, Homotopy and Applications

## Volume 10 (2008)

### Number 1

### On the homotopy groups of toric spaces

Pages: 437 – 479

DOI: http://dx.doi.org/10.4310/HHA.2008.v10.n1.a19

#### Author

#### Abstract

Given a certain class of simple polyhedral complexes $P$ and the associated Borel space $B_TP$ we compute the $E_2$-term of the Unstable Adams Novikov Spectral Sequence for $B_TP$ through a range. As a result, through a range, the higher homotopy groups of $B_TP$ 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 $M^{2n}(λ)$ can be computed through a range that was previously unknown. As an application, the homotopy type of a family of moment angle complexes $Z_K$ will be determined.

#### Keywords

homotopy group, quasi-toric manifold, subspace arrangement complement

#### 2010 Mathematics Subject Classification

Primary 47A15. Secondary 46A32, 47Dxx.