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.