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Sectional category with respect to group actions and sequential topological complexity of fibre bundles

Ramandeep Singh Arora, Navnath Daundkar, Soumen Sarkar

TL;DR

The paper develops a framework of sectional category with respect to a group action and introduces equivariant invariants $\mathrm{cat}_G^{\#}(X)$, $\mathrm{TC}_{k,G}^{\#}(X)$, and $\mathrm{TC}_{k,G}^{\#,*}(X)$, establishing their relationships to classical LS-category and sequential topological complexity. It proves key properties, including fibre-homotopy and product-type inequalities, and derives additive upper bounds for the sequential TC of total spaces of fibre bundles in terms of base motion planners and fibre invariants, such as $\mathrm{TC}_k(E) \le m + \mathrm{TC}_{k,G}^{\#,*}(F) - 1$. The authors apply these results to bound and compute invariants for generalized projective product spaces and mapping tori, using cohomological tools like cup-length and $\mathrm{zcl}_k$ to obtain sharp lower bounds and concrete values in several cases, including spheres and circles under group actions. Overall, the work unifies equivariant and non-equivariant perspectives on sectional-type invariants, providing concrete, computable bounds for complex fibre-bundle constructions with broad applications in topology and geometry.

Abstract

Let $X$ be a $G$-space. In this paper, we introduce the notion of sectional category with respect to $G$. As a result, we obtain $G$-homotopy invariants: the LS category with respect to $G$, the sequential topological complexity with respect to $G$ (which is same as the weak sequential equivariant topological complexity $\mathrm{TC}_{k,G}^w(X)$ in the sense of Farber and Oprea), and the strong sequential topological complexity with respect to $G$, denoted by $\mathrm{cat}_G^{\#}(X)$, $\mathrm{TC}_{k,G}^{\#}(X)$, and $\mathrm{TC}_{k,G}^{\#,*}(X)$, respectively. We explore several relationships among these invariants and well-known ones, such as the LS category, the sequential (equivariant) topological complexity, and the sequential strong equivariant topological complexity. In one of our main results, we give an additive upper bound for $\mathrm{TC}_k(E)$ for a fibre bundle $F \hookrightarrow E \to B$ with structure group $G$ in terms of certain motion planning covers of the base $B$ and the invariant $\mathrm{TC}_{k,G}^{\#,*}(F)$ or $\mathrm{cat}_{G^k}^{\#}(F^k)$, where the fibre $F$ is viewed as a $G$-space. As applications of these results, we give bounds on the sequential topological complexity of generalized projective product spaces and mapping tori.

Sectional category with respect to group actions and sequential topological complexity of fibre bundles

TL;DR

The paper develops a framework of sectional category with respect to a group action and introduces equivariant invariants , , and , establishing their relationships to classical LS-category and sequential topological complexity. It proves key properties, including fibre-homotopy and product-type inequalities, and derives additive upper bounds for the sequential TC of total spaces of fibre bundles in terms of base motion planners and fibre invariants, such as . The authors apply these results to bound and compute invariants for generalized projective product spaces and mapping tori, using cohomological tools like cup-length and to obtain sharp lower bounds and concrete values in several cases, including spheres and circles under group actions. Overall, the work unifies equivariant and non-equivariant perspectives on sectional-type invariants, providing concrete, computable bounds for complex fibre-bundle constructions with broad applications in topology and geometry.

Abstract

Let be a -space. In this paper, we introduce the notion of sectional category with respect to . As a result, we obtain -homotopy invariants: the LS category with respect to , the sequential topological complexity with respect to (which is same as the weak sequential equivariant topological complexity in the sense of Farber and Oprea), and the strong sequential topological complexity with respect to , denoted by , , and , respectively. We explore several relationships among these invariants and well-known ones, such as the LS category, the sequential (equivariant) topological complexity, and the sequential strong equivariant topological complexity. In one of our main results, we give an additive upper bound for for a fibre bundle with structure group in terms of certain motion planning covers of the base and the invariant or , where the fibre is viewed as a -space. As applications of these results, we give bounds on the sequential topological complexity of generalized projective product spaces and mapping tori.
Paper Structure (10 sections, 46 theorems, 102 equations)

This paper contains 10 sections, 46 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sectional category with respect to group actions and induced invariants
  4. Sectional category with respect to $G$
  5. LS category with respect to $G$
  6. Sequential topological complexity and strong sequential topological complexity with respect to $G$
  7. Sequential topological complexity of fibre bundles
  8. LS category of fibre bundles
  9. Application to generalized projective product spaces
  10. Application to mapping tori

Key Result

Theorem 1.1

Let $F \hookrightarrow E \xrightarrow{p} B$ be a fibre bundle with structure group $G$ where $E^k$ is a completely normal space. Let $\{U_1, \dots, U_m\}$ be an open cover of $B^k$ with sequential motion planners ${s_i \colon U_i \to B^I}$. If there exists a closed cover $\{R_1, \dots, R_m\}$ of $B$ In particular, $\mathrm{TC}_k(E) \leq m+\mathrm{cat}_{G^k}^{\#}(F^k)-1$ and $\mathrm{TC}_k(E) \leq

Theorems & Definitions (105)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Svasecat
  • Proposition 2.2: CLOT
  • Proposition 2.3: CLOT
  • Definition 2.4: Fadelleqcat
  • Proposition 2.5: RUD2010
  • Definition 3.1
  • Proposition 3.2
  • ...and 95 more