On homotopy types of Euclidean Rips complexes

TL;DR

This paper investigates the topological properties of Euclidean Rips complexes, showing their fundamental group behavior, universality in modeling homotopy types, and implications for group realizations and geometric structures.

Contribution

It proves the surjectivity of the fundamental group projection in R^3, establishes universality of Rips complexes in R^n, and explores geometric constraints in R^2.

Findings

Projection map induces a surjection on fundamental groups in R^3

Rips complexes model all homotopy types of PL-embeddable complexes in R^n

Any finitely presented group appears as a fundamental group of a Rips complex in R^4

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.