上极限、Stanley-Reisner代数和闭路空间
Colimits, Stanley-Reisner algebras, and loop spaces
摘要：
我们研究与有限的单纯复形K，在各种代数和拓扑类别的图。我们将它们的共极限与代数、组合学、几何和拓扑学中的熟悉结构联系起来。其中包括：直角Artin和Coxeter群（以及它们的复杂类似物，我们称之为循环群）; Stanley-Reisner代数和余代数; Davis和Januszkiewicz的空间DJ（K）与环面流形及其推广;和协调子空间安排。当K是一个标志复形时，我们通过确认相关的循环群同伦等价于循环空间$ΩDJ(K)$，推广了Artin和Coxeter群上的著名结果。定义了拓扑幺半群和拓扑群图的同伦余极限，并证明了它们在适当推广的意义上与分类空间的形成是可交换的。我们证明了拓扑群的相应图的同伦上极限是任意复K的$ΩDJ(K)$的一个模型，并且当K是flag时，到原上极限的自然投影是一个同伦等价.在这种情况下，这两种模式是兼容的。
英文摘要：
We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: right-angled Artin and Coxeter groups (and their complex analogues, which we call circulation groups); Stanley-Reisner algebras and coalgebras; Davis and Januszkiewicz's spaces DJ(K) associated with toric manifolds and their generalisations; and coordinate subspace arrangements. When K is a flag complex, we extend well-known results on Artin and Coxeter groups by confirming that the relevant circulation group is homotopy equivalent to the space of loops $ΩDJ(K)$. We define homotopy colimits for diagrams of topological monoids and topological groups, and show they commute with the formation of classifying spaces in a suitably generalised sense. We deduce that the homotopy colimit of the appropriate diagram of topological groups is a model for $ΩDJ(K)$ for an arbitrary complex K, and that the natural projection onto the original colimit is a homotopy equivalence when K is flag. In this case, the two models are compatible.
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