Contractibility of the Rips complexes of Integer lattices via local domination

TL;DR

This paper proves that Rips complexes of integer lattices in any dimension are contractible at sufficiently large scales, using a new local geometric concept called local crushing, with verified cases for dimensions 1 to 3.

Contribution

Introduction of the concept of locally dominated vertices and local crushing to establish contractibility of Rips complexes in integer lattices.

Findings

Rips complexes of integer lattices are contractible above certain scales.

Contractibility bounds are derived from Jung's constants.

Confirmed contractibility for dimensions 1, 2, and 3.

Abstract

We prove that for each positive integer $n$, the Rips complexes of the $n$-dimensional integer lattice in the $d_1$ metric (i.e., the Manhattan metric, also called the natural word metric in the Cayley graph) are contractible at scales above $n^2(2n-1)$, with the bounds arising from the Jung's constants. We introduce a new concept of locally dominated vertices in a simplicial complex, upon which our proof strategy is based. This allows us to deduce the contractibility of the Rips complexes from a local geometric condition called local crushing. In the case of the integer lattices in dimension $n$ and a fixed scale $r$, this condition entails the comparison of finitely many distances to conclude that the corresponding Rips complex is contractible. In particular, we are able to verify that for $n=1,2,3$, the Rips complex of the $n$-dimensional integer lattice at scale greater or equal to $n$ is contractible. We conjecture that the same proof strategy can be used to extend this result to all dimensions $n$