On Homotopy Types of Euclidean Rips Complexes

  title={On Homotopy Types of Euclidean Rips Complexes},
  author={Michal Adamaszek and F. Frick and A. Vakili},
  journal={Discrete & Computational Geometry},
  • Michal Adamaszek, F. Frick, A. Vakili
  • Published 2017
  • Mathematics, Computer Science
  • Discrete & Computational Geometry
  • The Rips complex at scale r of a set of points X in a metric space is the abstract simplicial complex whose faces are determined by finite subsets of X of diameter less than r. We prove that for X in the Euclidean 3-space $$\mathbb {R}^3$$R3 the natural projection map from the Rips complex of X to its shadow in $$\mathbb {R}^3$$R3 induces a surjection on fundamental groups. This partially answers a question of Chambers, de Silva, Erickson and Ghrist who studied this projection for subsets of… CONTINUE READING

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    Publications referenced by this paper.
    Vietoris–Rips Complexes of Planar Point Sets
    • 49
    • Highly Influential
    • PDF
    Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold
    • 63
    • Highly Influential
    • PDF
    Persistence stability for geometric complexes
    • 146
    • PDF
    Combinatorial Algebraic Topology
    • 446
    • PDF
    On the Vietoris-Rips complexes and a Cohomology Theory for metric spaces
    • 105
    • Highly Influential
    Algebraic Topology. EMS Textbooks in Mathematics
    • 2008
    Topological methods
    • 567