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 3space 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 PLembeddable 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2016
 arXiv:
 arXiv:1602.04131
 Bibcode:
 2016arXiv160204131A
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Metric Geometry
 EPrint:
 Discrete &