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

Ramandeep Singh Arora, Sutirtha Datta, Navnath Daundkar, Gopal Chandra Dutta

TL;DR

This work introduces the $m$-dimensional sectional category $\mathrm{secat}_m(p)$ as a computable, finite-dimensional proxy for Schwarz's sectional category and develops a cohesive framework of induced invariants: the $m$-category $\mathrm{cat}_m$, the $m$-topological complexity $\mathrm{TC}^m$, and the $m$-homotopic distance $\mathrm{D}_m$ between maps. It establishes invariance properties, product and lower/upper bounds, and cohomological techniques (including the $m$-cohomological distance) that refine classical bounds; it also provides explicit computations for Moore spaces, products of spheres, complex and real projective spaces, surfaces, and spaces with $H$-space structures, illustrating stabilization phenomena and when $m$-dimensional invariants coincide with their classical counterparts. A unifying thread is the translation of $m$-dimensional data into fibrational or pullback constructions, enabling express connections such as $\mathrm{TC}^m(X)=\mathrm{D}_m(pr_1,pr_2)$ and, for $H$-spaces, $\mathrm{TC}^m(X)=\mathrm{cat}_m(X)$. The paper also links $m$-cohomological distance to $m$-homotopic distance, yielding sharp lower bounds and dimension-based estimates that sharpen traditional genus-type arguments. Collectively, the results provide a practical, scalable toolkit for estimating motion-planning-type invariants and understanding their stabilization across dimensions in homotopy theory and related applications.

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.

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

TL;DR

This work introduces the -dimensional sectional category as a computable, finite-dimensional proxy for Schwarz's sectional category and develops a cohesive framework of induced invariants: the -category , the -topological complexity , and the -homotopic distance between maps. It establishes invariance properties, product and lower/upper bounds, and cohomological techniques (including the -cohomological distance) that refine classical bounds; it also provides explicit computations for Moore spaces, products of spheres, complex and real projective spaces, surfaces, and spaces with -space structures, illustrating stabilization phenomena and when -dimensional invariants coincide with their classical counterparts. A unifying thread is the translation of -dimensional data into fibrational or pullback constructions, enabling express connections such as and, for -spaces, . The paper also links -cohomological distance to -homotopic distance, yielding sharp lower bounds and dimension-based estimates that sharpen traditional genus-type arguments. Collectively, the results provide a practical, scalable toolkit for estimating motion-planning-type invariants and understanding their stabilization across dimensions in homotopy theory and related applications.

Abstract

In this paper, we systematically study the -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 -dimensional Lusternik-Schnirelmann category, the -topological complexity, and the -homotopic distance between maps. We further investigate the relationships between these -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 -cohomological distance and study its interaction with the -homotopic distance.
Paper Structure (21 sections, 64 theorems, 124 equations)

This paper contains 21 sections, 64 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. $m$-sectional category and $m$-category
  3. $m$-homotopic distance and $m$-topological complexity
  4. Notations and Conventions
  5. $m$-sectional category
  6. Bounds on $m$-sectional category
  7. Normal spaces and product inequality
  8. $m$-category
  9. $m$-category of a map
  10. Bounds on $m$-category
  11. $m$-homotopic distance
  12. Properties
  13. Invariance
  14. Bounds on $m$-homotopic distance
  15. $m$-Cohomological distance
  16. ...and 6 more sections

Key Result

Proposition 2.3

Suppose $p \colon E \to B$ is a fibration. Then an open subset of $B$ satisfies property $A_{m,\,p}$ if and only if it satisfies the property $A_{m,\,p}'$. In particular,

Theorems & Definitions (130)

  • Definition 2.1
  • Definition 2.2: schwarz1961genus
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 120 more