An Upper Bound for the Depth of Some Classes of Polyhedra

TL;DR

This paper introduces new concepts of capacity and depth in category theory, computes these for certain groups, and establishes an upper bound for the depth of specific classes of finite polyhedra, extending previous results.

Contribution

It generalizes the notion of depth to arbitrary categories, computes these invariants for groups, and provides an upper bound for polyhedra depth, advancing understanding in algebraic topology.

Findings

Computed capacity and depth for certain groups

Established an upper bound for polyhedra depth

Generalized previous results in polyhedral topology

Abstract

K. Borsuk in the seventies introduced the notions of capacity and depth of compacta together with some relevant problems. In this paper, first, we introduce the concepts of the (strong) capacity and the (strong) depth of an object in an arbitrary category. Then in the category of groups, we compute the (strong) capacity and the (strong) depth of some well-known groups. Finally, we find an upper bound for the depth of some classes of finite polyhedra which generalizes a result of D. Kolodziejczyk in this subject.