标题： 关于群作用的截面范畴与纤维丛的序列拓扑复杂度 显示英文标题

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

摘要： 设 $X$ 为一个 $G$-空间。 本文中，我们引入了相对于 $G$ 的 sectional category 的概念。 因此，我们得到$G$-同伦不变量：相对于$G$的LS范畴，相对于$G$的序列拓扑复杂度（与Farber和Oprea意义上的弱序列等变拓扑复杂度$\mathrm{TC}_{k,G}^w(X)$相同），以及相对于$G$的强序列拓扑复杂度，分别记为$\mathrm{cat}_G^{\#}(X)$、$\mathrm{TC}_{k,G}^{\#}(X)$和$\mathrm{TC}_{k,G}^{\#,*}(X)$。 我们探讨了这些不变量与著名的不变量（如LS范畴、序列（等变）拓扑复杂度以及序列强等变拓扑复杂度）之间的若干关系。 在我们的主要结果之一中，我们给出了一个纤维丛 $F \hookrightarrow E \to B$ 的 $\mathrm{TC}_k(E)$ 的加法上界，其中结构群为 $G$，以基空间 $B$ 的某些运动规划覆盖和不变量 $\mathrm{TC}_{k,G}^{\#,*}(F)$ 或 $\mathrm{cat}_{G^k}^{\#}(F^k)$ 表示，纤维 $F$ 被视为 $G$-空间。 作为这些结果的应用，我们给出了广义射影乘积空间和映射环面的序列化拓扑复杂度的界。 显示英文摘要

摘要： 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.