Abstract
We study diagrams associated with a finite simplicial complex Kin 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 Ω DJ(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 Ω DJ(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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Panov, T., Ray, N., Vogt, R. (2003). Colimits, Stanley-Reisner Algebras, and Loop Spaces. In: Arone, G., Hubbuck, J., Levi, R., Weiss, M. (eds) Categorical Decomposition Techniques in Algebraic Topology. Progress in Mathematics, vol 215. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7863-0_15
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DOI: https://doi.org/10.1007/978-3-0348-7863-0_15
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