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.
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机译:给定一类简单的多面体复合体$ P $和相关的Borel空间$ B_TP $,我们计算了整个范围内$ B_TP $的不稳定亚当斯诺维科夫光谱序列的$ E_2 $项。结果,在一定范围内,$ B_TP $的较高同伦基团与尺寸取决于$ P $的组合的球体楔形的同伦基团同构。本文提供了一种统一的方法,可以解决计算任意复杂坐标子空间排列的补码的较高同伦组的问题。在复杂坐标子空间排列的补码的同伦类型未知的情况下,我们扩展了所有更高的同伦组计算。如果$ K $是定义一个球的三角剖分的简单复形,该球对简单的凸多面体$ P $对偶,那么在许多情况下,准变矩流形$ M ^ {2n}(λ )$可以通过以前未知的范围进行计算。作为应用,将确定矩角复合体$ Z_K $的同伦类型。
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