On Upper Bounds for the Depth of some Classes of Polyhedra

TL;DR

This paper establishes upper bounds on the depth of various classes of polyhedra, including those with specific fundamental groups, and demonstrates the sharpness of some bounds through examples.

Contribution

It introduces new upper bounds for the depth of certain polyhedra classes, expanding understanding of their topological complexity.

Findings

Derived upper bounds for polyhedra with finite fundamental group

Provided bounds for polyhedra with abelian or free fundamental groups

Examples showing some bounds are sharp

Abstract

In this paper, we present upper bounds for the depth of some classes of polyhedra, including: polyhedra with finite fundamental group, polyhedra $P$ with abelian or free $\pi_1(P)$ and finitely generated $H_i(tilde{P};\mathbb{Z}$, 2-dimensional polyhedra with abelian or free fundamental group, and 2-dimensional polyhedra with elementary amenable fundamental group $G$ with finite cohomological dimension $cd(G)$. Furthermore, we provide some examples to show that some of these bounds are sharp.