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

TL;DR

This paper introduces new $G$-equivariant invariants for $G$-spaces, explores their relationships with existing invariants, and provides bounds on the sequential topological complexity of fibre bundles and related spaces.

Contribution

It defines the sectional category with respect to a group action and develops $G$-homotopy invariants, establishing bounds and relationships among these invariants and classical topological complexity measures.

Findings

Introduces $G$-equivariant sectional category and related invariants.

Provides an additive upper bound for the topological complexity of fibre bundles.

Applies results to bounds on complexity of projective product spaces and mapping tori.

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.