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 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.