Homology of Spaces of Directed Paths in Euclidean Pattern Spaces

  • Roy Meshulam
  • Martin Raussen


Let \(\mathcal{F}\) be a family of subsets of {1, , n} and let
$$\displaystyle{ Y _{\mathcal{F}} =\bigcup _{F\in \mathcal{F}}\{(x_{1},\ldots,x_{n}) \in \mathbb{R}^{n}: x_{ i} \in \mathbb{Z}\text{for all}i \in F\}. }$$
Let \(X_{\mathcal{F}} = \mathbb{R}^{n}\setminus Y _{\mathcal{F}}\). For a vector of positive integers k = (k1, , k n ) let \(\vec{P}(X_{\mathcal{F}})_{\mathbf{0}}^{\mathbf{k}+\mathbf{1}}\) denote the space of monotone paths from 0 = (0, , 0) to k + 1 = (k1 + 1, , k n + 1) whose interior is contained in \(X_{\mathcal{F}}\). The path spaces \(\vec{P}(X_{\mathcal{F}})_{\mathbf{0}}^{\mathbf{k}+\mathbf{1}}\) appear as natural examples in the study of Dijkstra’s PV-model for parallel computations in concurrency theory.

We study the topology of \(\vec{P}(X_{\mathcal{F}})_{\mathbf{0}}^{\mathbf{k}+\mathbf{1}}\) by relating it to a subspace arrangement in a product of simplices. This, in particular, leads to a computation of the homology of \(\vec{P}(X_{\mathcal{F}})_{\mathbf{0}}^{\mathbf{k}+\mathbf{1}}\) in terms of certain order complexes associated with the hypergraph \(\mathcal{F}\).


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© Springer International publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsTechnionHaifaIsrael
  2. 2.Department of Mathematical SciencesAalborg UniversityAalborg ØstDenmark

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