Linear embeddings of contractible and collapsible complexes

TL;DR

This paper explores the complexity of embedding contractible and collapsible complexes in Euclidean space, linking algebraic properties of group presentations to geometric embedding difficulty, and establishing bounds on embedding after subdivisions.

Contribution

It demonstrates a connection between the complexity of trivializing group presentations and embedding complexity of their complexes, and provides optimal bounds for embedding collapsible complexes.

Findings

Hard-to-trivialize presentations lead to complexes requiring many subdivisions to embed.

All collapsible d-complexes with n facets embed in $\\mathbb{R}^{2d}$ after fewer than n barycentric subdivisions.

Cones over non-planar graphs do not embed in $\\mathbb{R}^{3}$, showing the bounds are tight.

Abstract

(1) We show that if a presentation of the trivial group is "hard to trivialize", in the sense that lots of Tietze moves are necessary to transform it into the trivial presentation, then the associated presentation complex (which is a contractible 2-dimensional cell complex) is "hard to embed in $\mathbb{R}^3$", in the sense that lots of linear subdivisions are necessary. (2) For any d, we show that all collapsible d-complexes with n facets linearly embed in $\mathbb{R}^{2d}$ after less than n barycentric subdivisions. This is best possible, as cones over non-planar graphs do not topologically embed in $\mathbb{R}^{3}$.