On Davis-Januszkiewicz homotopy types I; Formality and rationalisation
On Davis-Januszkiewicz homotopy types I; Formality and rationalisation
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关于 Davis-Januszkiewicz 同伦类型 I；
DOI：
10.2140/agt.2005.5.31
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发表时间：
2003
影响因子：
0.7
通讯作者：
N. Ray
中科院分区：
文献类型：
--
作者：
D. Notbohm;N. Ray
对于任意简单复K， Davis和Januszkiewicz定义了一类同伦等价的cw -复，它们的整上同调环与K的Stanley-Reisner代数同构。随后，Buchstaber和Panov给出了另一种构造（这里称为c(K)），他们证明了它与Davis和Januszkiewicz的例子是同伦等价的。因此，研究一个空间的同伦类型在多大程度上是由这样一个上同伦环决定的是很自然的。我们从这里开始研究，在模型范畴理论的背景下。特别地，我们推广了Franz的工作，证明了c(K)的奇异协链代数的形式是微分渐变非交换代数。我们通过证明Sullivan's可交换协链代数的相应结果来专门研究理数，并推导出c(K)的理数对于特殊的复合体K族是唯一的。在后续中，我们将分别考虑c(K)在每个素数处的唯一性，并应用Sullivan's等差平方来产生该族的全局结果。AMS分类55P62、55U05；05年e99
For an arbitrary simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose inte- gral cohomology rings are isomorphic to the Stanley-Reisner algebra of K. Subsequently, Buchstaber and Panov gave an alternative construction (here called c(K)), which they showed to be homotopy equivalent to Davis and Januszkiewicz's examples. It is therefore natural to investigate the extent to which the homotopy type of a space is determined by having such a co- homology ring. We begin this study here, in the context of model category theory. In particular, we extend work of Franz by showing that the singular cochain algebra of c(K) is formal as a differential graded noncommutative algebra. We specialise to the rationals by proving the corresponding result for Sullivan's commutative cochain algebra, and deduce that the rationali- sation of c(K) is unique for a special family of complexes K. In a sequel, we will consider the uniqueness of c(K) at each prime separately, and apply Sullivan's arithmetic square to produce global results for this family. AMS Classification 55P62, 55U05; 05E99