Download this article
 Download this article For screen
For printing
Recent Issues

Volume 24
Issue 3, 1225–1808
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
 
Other MSP Journals
Sequential parametrized topological complexity and related invariants

Michael Farber and John Oprea

Algebraic & Geometric Topology 24 (2024) 1755–1780
Abstract

Parametrized motion planning algorithms have a high degree of universality and flexibility; they generate the motion of a robotic system under a variety of external conditions. The latter are viewed as parameters and constitute part of the input of the algorithm. The concept of sequential parametrized topological complexity TCr[p: E B] is a measure of the complexity of such algorithms. It was studied by Cohen, Farber and Weinberger (2021, 2022) for r = 2 and by Farber and Paul (2022) for r 2. We analyze the dependence of the complexity TCr[p: E B] on an initial bundle with structure group G and on its fibre X viewed as a G–space. Our main results estimate TCr[p: E B] in terms of certain invariants of the bundle and the action on the fibre. Moreover, we also obtain estimates depending on the base and the fibre. Finally, we develop a calculus of sectional categories featuring a new invariant secatf[p: E B] which plays an important role in the study of sectional category of towers of fibrations.

Keywords
topological complexity, sectional category, fibration
Mathematical Subject Classification
Primary: 55M30
References
Publication
Received: 5 September 2022
Revised: 6 January 2023
Accepted: 16 January 2023
Published: 28 June 2024
Authors
Michael Farber
School of Mathematical Sciences
Queen Mary University of London
London
United Kingdom
John Oprea
Department of Mathematics
Cleveland State University
Cleveland, OH
United States

Open Access made possible by participating institutions via Subscribe to Open.