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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Monotonicity of the Schwarz genus
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by Petar Pavešić PDF
Proc. Amer. Math. Soc. 148 (2020), 1339-1349 Request permission

Abstract:

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.
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Additional Information
  • Petar Pavešić
  • Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia
  • Email: petar.pavesic@fmf.uni-lj.si
  • Received by editor(s): December 21, 2018
  • Received by editor(s) in revised form: July 8, 2019
  • Published electronically: October 28, 2019
  • Additional Notes: The author was supported by the Slovenian Research Agency research grant P1-0292 and research project J1-7025
  • Communicated by: Mark Behrens
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 1339-1349
  • MSC (2010): Primary 55M30, 55S40
  • DOI: https://doi.org/10.1090/proc/14791
  • MathSciNet review: 4055959