Homology of Spaces of Directed Paths in Euclidean Pattern Spaces



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 \(\mathcal{F}\).


  1. 1.
    V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko, Singularities of Differentiable Maps, vol. 1 (Birkhäuser, Basel, 1985)CrossRefMATHGoogle Scholar
  2. 2.
    A. Björner, M.L. Wachs, Shellable nonpure complexes and posets. I. Trans. Am. Math. Soc. 348, 1299–1327 (1996)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    A. Björner, V. Welker, The homology of “k-equal” manifolds and related partition lattices. Adv. Math. 110, 277–313 (1995)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    E.W. Dijkstra, Co-operating sequential processes, in Programming Languages, ed. by F. Genuys (Academic, New York, 1968), pp. 43–110Google Scholar
  5. 5.
    L. Fajstrup, É. Goubault, M. Raussen, Algebraic topology and concurrency. Theor. Comput. Sci. 357, 241–278 (2006). Revised version of Aalborg University (1999, preprint)Google Scholar
  6. 6.
    L. Fajstrup, É. Goubault, E. Haucourt, S. Mimram, M. Raussen, Directed Algebraic Topology and Concurrency (Springer, Berlin, 2016)CrossRefMATHGoogle Scholar
  7. 7.
    M. Goresky, R. MacPherson, Stratified Morse Theory (Springer, Berlin, 1988)CrossRefMATHGoogle Scholar
  8. 8.
    M. Hirsch, Differential Topology (Springer, New York, 1976)CrossRefMATHGoogle Scholar
  9. 9.
    I. Peeva, V. Reiner, V. Welker, Cohomology of real diagonal subspace arrangements via resolutions. Compositio Math. 117, 99–115 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    V. Pratt, Modelling concurrency with geometry, in Proceedings of the 18th ACM Symposium on Principles of Programming Languages (1991), pp. 311–322Google Scholar
  11. 11.
    M. Raussen, Simplicial models for trace spaces. Algebr. Geom. Topol. 10, 1683–1714 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    M. Raussen, Simplicial models for trace spaces II: general higher-dimensional automata. Algebr. Geom. Topol. 12, 1745–1765 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    M. Raussen, K. Ziemiański, Homology of spaces of directed paths on Euclidean cubical complexes. J. Homotopy Relat. Struct. 9, 67–84 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    R.P. Stanley, Acyclic orientations of graphs. Discrete Math. 5, 171–178 (1973)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    R. van Glabbeek, On the Expressiveness of higher dimensional automata. Theor. Comput. Sci. 368, 168–194 (2006)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    J.W. Walker, Canonical homeomorphisms of posets. Eur. J. Combin. 9, 97–107 (1988)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    G. Ziegler, R. Živaljević, Homotopy types of subspace arrangements via diagrams of spaces. Math. Ann. 295, 527–548 (1993)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    K. Ziemiański, On execution spaces of PV-programs. Theoret. Comput. Sci. 619, 87–98 (2016)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International publishing AG 2017

Authors and Affiliations

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

Personalised recommendations