Sequential parametrized topological complexity of sphere bundles

Michael Farber*, Amit Kumar Paul

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Autonomous motion of a system (robot) is controlled by a motion planning algorithm. A sequential parametrized motion planning algorithm [10] works under variable external conditions and generates continuous motions of the system to attain the prescribed sequence of states at prescribed moments of time. Topological complexity of such algorithms measures their structure and discontinuities. Information about states of the system consistent with states of the external conditions is described by a fibration p:E→B where the base B parametrizes the external conditions and each fibre p-1(b) is the configuration space of the system constrained by external conditions b∈B; more detail on this approach is given below. Our main goal in this paper is to study the sequential topological complexity of sphere bundles ξ˙:E˙→B; in other words we study “parametrized families of spheres” and sequential parametrized motion planning algorithms for such bundles. We use the Euler and Stiefel–Whitney characteristic classes to obtain lower bounds on the topological complexity. We illustrate our results by many explicit examples. Some related results for the special case r=2 were described earlier in [13].

Original languageEnglish
Article number53
JournalEuropean Journal of Mathematics
Volume10
Issue number4
DOIs
StatePublished - Dec 2024
Externally publishedYes

Funding

FundersFunder number
Engineering and Physical Sciences Research Council

    Keywords

    • 55M30
    • Parametrized topological complexity
    • Sequential parametrized topological complexity
    • Sphere bundle
    • Topological complexity

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