On the $m$-dimensional sectional category and induced invariants

TL;DR

This paper introduces and develops the theory of the $m$-dimensional sectional category of a fibration, creating a hierarchy of invariants that generalize classical concepts like Lusternik-Schnirelmann category and topological complexity.

Contribution

It systematically studies the $m$-dimensional sectional category, establishing its properties and relationships with classical invariants, and introduces new concepts like $m$-cohomological distance.

Findings

The $m$-dimensional sectional category provides a hierarchy of invariants.

Examples show when $m$-invariants agree or differ from classical ones.

Interactions between $m$-cohomological distance and $m$-homotopic distance are analyzed.

Abstract

In this paper, we systematically study the $m$-dimensional sectional category of a fibration, introduced by Schwarz, as an approximating invariant for the sectional category. We develop the basic theory of this invariant, establish its fundamental properties, and show how it gives rise to a hierarchy of induced invariants, including the $m$-dimensional Lusternik-Schnirelmann category, the $m$-topological complexity, and the $m$-homotopic distance between maps. We further investigate the relationships between these $m$-dimensional invariants and their classical analogues, present a variety of examples in which these invariants are computed, and illustrate when they agree with or differ from their classical counterparts. We also introduce the notion of $m$-cohomological distance and study its interaction with the $m$-homotopic distance.