## Monotonicity of the Schwarz genus

HTML articles powered by AMS MathViewer

- 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.

## References

- Octavian Cornea,
*Lusternik-Schnirelmann-categorical sections*, Ann. Sci. École Norm. Sup. (4)**28**(1995), no. 6, 689–704. MR**1355138** - Octav Cornea, Gregory Lupton, John Oprea, and Daniel Tanré,
*Lusternik-Schnirelmann category*, Mathematical Surveys and Monographs, vol. 103, American Mathematical Society, Providence, RI, 2003. MR**1990857**, DOI 10.1090/surv/103 - Michael Farber,
*Invitation to topological robotics*, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR**2455573**, DOI 10.4171/054 - Yves Felix, Steve Halperin, and Jean-Claude Thomas,
*Lusternik-Schnirelmann category of skeleta*, Topology Appl.**125**(2002), no. 2, 357–361. MR**1933583**, DOI 10.1016/S0166-8641(01)00288-7 - Aleksandra Franc and Petar Pavešić,
*Spaces with high topological complexity*, Proc. Roy. Soc. Edinburgh Sect. A**144**(2014), no. 4, 761–773. MR**3233755**, DOI 10.1017/S030821051200087X - Mark Grant, Gregory Lupton, and John Oprea,
*Spaces of topological complexity one*, Homology Homotopy Appl.**15**(2013), no. 2, 73–81. MR**3117387**, DOI 10.4310/HHA.2013.v15.n2.a4 - Petar Pavešić,
*Complexity of the forward kinematic map*, Mechanism and Machine Theory**117**(2017), 230-243. - Petar Pavešić,
*Topological complexity of a map*, Homology Homotopy Appl.**21**(2019), no. 2, 107–130. MR**3921612**, DOI 10.4310/HHA.2019.v21.n2.a7 - Petar Pavešić and Renzo A. Piccinini,
*Fibrations and their classification*, Research and Exposition in Mathematics, vol. 33, Heldermann Verlag, Lemgo, 2013. MR**3059013** - Albert S. Schwarz,
*The genus of a fiber space*, Amer. Math. Soc. Transl. (2)**55**(1966), 49–140. - Edwin H. Spanier,
*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112**

## 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