# On Homotopy Types of Euclidean Rips Complexes

- 112 Downloads

## 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.

## Keywords

Vietoris–Rips complex Shadow Fundamental group Homotopy## Mathematics Subject Classification

05E45 55U10 52C99## Notes

### 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.

## References

- 1.Attali, D., Lieutier, A., Salinas, D.: Vietoris–Rips complexes also provide topologically correct reconstructions of sampled shapes. Comput. Geom.
**46**(4), 448–465 (2013)Google Scholar - 2.Björner, A.: Topological methods. In: Graham, R., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, vol. 2, pp. 1819–1872. Elsevier, Amsterdam (1995)Google Scholar
- 3.Chambers, E.W., de Silva, V., Erickson, J., Ghrist, R.: Vietoris–Rips complexes of planar point sets. Discrete Comput. Geom.
**44**(1), 75–90 (2010)MathSciNetCrossRefMATHGoogle Scholar - 4.Chazal, F., de Silva, V., Oudot, S.: Persistence stability for geometric complexes. Geom. Dedicata
**173**, 193–214 (2014)MathSciNetCrossRefMATHGoogle Scholar - 5.Deza, M., Dutour, M., Shtogrin, M.: On simplicial and cubical complexes with short links. Isr. J. Math.
**144**(1), 109–124 (2004)MathSciNetCrossRefMATHGoogle Scholar - 6.Dranišnikov, A.N., Repovš, D.: Embedding up to homotopy type in Euclidean space. Bull. Aust. Math. Soc.
**47**(1), 145–148 (1993)MathSciNetCrossRefMATHGoogle Scholar - 7.Hausmann, J.-C.: On the Vietoris–Rips complexes and a cohomology theory for metric spaces. In: Quinn, W. (ed.) Prospects in Topology. Annals of Mathematics Studies, vol. 138, pp. 175–188. Princeton University Press, Princeton (1995)Google Scholar
- 8.Kozlov, D.N.: Combinatorial Algebraic Topology. Algorithms and Computation in Mathematics, vol. 21. Springer, Berlin (2008)CrossRefGoogle Scholar
- 9.Latschev, J.: Vietoris–Rips complexes of metric spaces near a closed Riemannian manifold. Arch. Math.
**77**(6), 522–528 (2001)MathSciNetCrossRefMATHGoogle Scholar - 10.tom Dieck, T.: Algebraic Topology. EMS Textbooks in Mathematics. European Mathematical Society, Zürich (2008)CrossRefGoogle Scholar
- 11.Vietoris, L.: Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen. Math. Ann.
**97**(1), 454–472 (1927)MathSciNetCrossRefMATHGoogle Scholar