施瓦茨亏格的单调性

摘要：

定义纤维化$ξ\colon E\to B$的Schwarz亏格$\mathsf{g}(ξ)$为最小整数$n$，使得存在$B$被允许$ξ$部分可截的$n$开集覆盖.许多重要的概念，包括Lusternik-Schnirelmann范畴、Farber的拓扑复杂性和Smale-Vassiliev的算法复杂性，都可以自然地表示为适当选择的纤维化的Schwarz亏格。本文研究了Schwarz亏格与纤维化之间的某些态射的关系。我们的主要结果是：如果存在纤维映射$ξ\colon E\to B$和$ξ'\colon E'\to B$之间的$f\colon E\to E'$诱导的$n$ -等价的各自的纤维之间的足够大的$n$，那么$\mathsf{g}(ξ)=\mathsf{g}(ξ')$。从这一点上，我们得出了几个有趣的结果，有关的拓扑复杂性的空间与拓扑复杂性，它的子空间（和类似的范畴）。例如，我们表明，如果一个CW-复合物具有高的拓扑复杂性（相对于其尺寸和连接性），那么它的拓扑复杂性，它的拓扑结构是一个增函数的尺寸。

英文摘要：

Schwarz genus $\mathsf{g}(ξ)$ of a fibration $ξ\colon E\to B$ is defined as the minimal integer $n$, such that there exists a cover of $B$ by $n$ open sets that admit partial section to $ξ$. Many important concepts, including Lusternik-Schnirelmann category, Farber's topological complexity and Smale-Vassiliev's complexity of algorithms can be naturally expressed as Schwarz genera of suitably chosen fibrations. In this paper we study Schwarz genus in relation with certain type of morphisms between fibrations. Our main result is the following: if there exist a fibrewise map $f\colon E\to E'$ between fibrations $ξ\colon E\to B$ and $ξ'\colon E'\to B$ which induces an $n$-equivalence between respective fibres for a sufficiently big $n$, then $\mathsf{g}(ξ)=\mathsf{g}(ξ')$. From this we derive several interesting results relating the topological complexity of a space with the topological complexities of its skeleta and subspaces (and similarly for the category). For example, we show that if a CW-complexes has high topological complexity (with respect to its dimension and connectivity), then the topological complexity of its skeleta is an increasing function of the dimension.

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