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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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Homology of Spaces of Directed Paths in Euclidean Pattern Spaces

Meshulam

Raussen

2017

A Journey Through Discrete Mathematics

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Let F be a family of subsets of {1, . . . , n} and letFor a vector of positive integers k = (k 1 , . . . , k n ) let P (X F ) k+1 0 denote the space of monotone paths from 0 = (0, . . . , 0) to k+ 1 = (k 1 + 1, . . . , k n + 1) whose interior is contained in X F . The path spaces P (X F ) k+1 0 appear as natural examples in the study of Dijkstra's PV-model for parallel computations in concurrency theory. We study the topology of P (X F ) k+1 0 by relating it to a subspace arrangement in a product of simplices. This, in particular, leads to a computation of the homology of P (X F ) k+1 0 in terms of certain order complexes associated with the hypergraph F.

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Cited by 21 publications

References 19 publications

Commutative Algebra of Subspace and Hyperplane Arrangements

Commutative Algebra of Subspace and Hyperplane Arrangements

Colimits, Stanley-Reisner Algebras, and Loop Spaces

Homology of Spaces of Directed Paths in Euclidean Pattern Spaces

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