20.500.12556/RUL-115220 Monotonicity of the Schwarz genus 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. 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. Schwarz genus Lusternik-Schnirelmann category sectional category topological complexity true false false Angleški jezik Slovenski jezik Neznano 2020-04-18 10:45:04 2020-04-18 10:45:07 2022-08-27 04:00:13 0000-00-00 00:00:00 2020 0 0 Str. 1339-1349 no. 3 Vol. 148 March 2020 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 515.14 0002-9939 10.1090/proc/14791 18784089 2335236 RAZ_Pavesic_Petar_2020.pdf RAZ_Pavesic_Petar_2020.pdf 1 E081CBD60CA126A0488965986F2D378B 83c91383dd52ee7e83d20791c00a3250cfa515537be0e8c5bd0ce7f682cbfbb5 6eab21c6-a1b8-11eb-a523-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=127917 Fakulteta za matematiko in fiziko 0 0 0