On homotopy types of Euclidean Rips complexes

Michal Adamaszek; Florian Frick; Adrien Vakili

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arXiv · 1602.04131

On homotopy types of Euclidean Rips complexes

Michal Adamaszek·Florian Frick·Adrien Vakili

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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 R^3 the natural projection map from the Rips complex of X to its shadow in 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 R^2. We further show that Rips complexes of finite subsets of R^n are universal, in that they model all homotopy types of simplicial complexes PL-embeddable in 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 R^4. We furthermore show that if the Rips complex of a finite point set in R^2 is a normal pseudomanifold of dimension at least two then it must be the boundary of a crosspolytope.

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Michal Adamaszek, Florian Frick, Adrien Vakili. 2016-02-12. On homotopy types of Euclidean Rips complexes. https://arxiv.org/abs/1602.04131

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