On Homotopy Types of Euclidean Rips Complexes

Abstract

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 \(\mathbb {R}^3\) the natural projection map from the Rips complex of X to its shadow in \(\mathbb {R}^3\) 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 \(\mathbb {R}^2\). We further show that Rips complexes of finite subsets of \(\mathbb {R}^n\) are universal, in that they model all homotopy types of simplicial complexes PL-embeddable in \(\mathbb {R}^n\). As an application we get that any finitely presented group appears as the fundamental group of a Rips complex of a finite subset of \(\mathbb {R}^4\). We furthermore show that if the Rips complex of a finite point set in \(\mathbb {R}^2\) is a normal pseudomanifold of dimension at least two then it must be the boundary of a crosspolytope.

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Notes

  1. 1.

    Chambers et al. [3] use the term k-connected to describe the situation when the induced map on \(\pi _k\) is also a bijection, although it is more standard to call the latter a k-equivalence.

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Acknowledgements

We thank Jesper M. Møller for helpful discussions and for suggesting the collaboration of the first and third author. We also thank the referees for their suggestions. Some of this research was performed while the second author visited the University of Copenhagen. The second author is grateful for the hospitality of the Department of Mathematical Sciences there. MA was supported by VILLUM FONDEN through the network for Experimental Mathematics in Number Theory, Operator Algebras, and Topology.

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Correspondence to Michał Adamaszek.

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Editor in Charge: Kenneth Clarkson

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Adamaszek, M., Frick, F. & Vakili, A. On Homotopy Types of Euclidean Rips Complexes. Discrete Comput Geom 58, 526–542 (2017). https://doi.org/10.1007/s00454-017-9916-5

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Keywords

  • Vietoris–Rips complex
  • Shadow
  • Fundamental group
  • Homotopy

Mathematics Subject Classification

  • 05E45
  • 55U10
  • 52C99