Monotonicity of the Schwarz genus

Pavešić, Petar (author)

The Schwarz genus $\mathsf{g}(\xi)$ of a fibration $\xi \colon E\to B$ is defined as the minimal integer $n$ such that there exists a cover of $B$ by $n$ open sets that admit partial sections to $\xi$. Many important concepts, including the 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 types of morphisms between fibrations. Our main result is the following: if there exists a fibrewise map $f\colon E\to E'$ between fibrations $\xi \colon E\to B$ and $\xi '\colon E'\to B$ which induces an $n$-equivalence between respective fibres for a sufficiently big $n$, then $\mathsf {g}(\xi )=\mathsf {g}(\xi ')$. 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-complex 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.

• Language: English
• Keywords: Schwarz genus, Lusternik-Schnirelmann category, sectional category, topological complexity
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2020
• Number of pages: Str. 1339-1349
• Numbering: Vol. 148, no. 3
• PID: 20.500.12556/RUL-115220
• UDC: 515.14
• ISSN on article: 0002-9939
• DOI: 10.1090/proc/14791
• COBISS.SI-ID: 18784089
• Publication date in RUL: 18.04.2020

PAVEŠIĆ, Petar, 2020, Monotonicity of the Schwarz genus. Proceedings of the American Mathematical Society [online]. 2020. Vol. 148, no. 3, p. 1339–1349. [Accessed 11 June 2025]. DOI 10.1090/proc/14791. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=115220

Record is a part of a journal

• Title: Proceedings of the American Mathematical Society
• Shortened title: Proc. Am. Math. Soc.
• Publisher: American Mathematical Society
• ISSN: 0002-9939
• COBISS.SI-ID: 2335236

Secondary language

• Language: Slovenian

Abstract

Švarcov rod $\mathsf{g}(\xi )$ vlaknenja $\xi\colon E\to B$ je minimalno celo število $n$, za katerega obstaja pokritje $B$ z $n$ odprtimim množicami, ki dopuščajo delni prerez za $\xi$. Veliko pomembnih pojmov je mogoče opisati s pomočjo Švarcovega roda primerno izbranega vlaknenja, npr. Lusternik-Schnirelmannovo kategorijo, Farberjevo topološko kompleksnost, Smale-Vassilievo kompleksnost algoritmov itn. V članku obravnavamo zvezo med Švarcovim rodom in določenim tipom vlakenskih morfizmov. Glavni rezultat pravi, da če obstaja vlakenska preslikava $f\colon E\to E'$ med vlaknenji $\xi\colon E\to B$ in $\xi'\colon E'\to B$, ki inducira $n$-ekvivalenco med pripadajočimi vlakni za dovolj velike $n$, potem je $\mathsf{g}(\xi)=\mathsf{g}(\xi')$. Od tod dobimo vrsto zanimivih primerjav med topološko kompleksnostjo prostora in topološkimi kompleksnostmi njegovih skeletov (ter podobno za LS-kategorijo). Za primer, pri CW kompleksih, ki imajo visoko topološko kompleksnost (glede na njihovo dimenzijo in povezanost) topološka kompleksnost skeletov narašča z dimenzijo.