Abstract:We study a probabilistic variant of the r-th sequential parametrized topological complexity, which bounds this classical invariant from below and measures the difficulty in constructing permissive parametrized motion planning algorithms. On one hand, we use cohomology to show that this new invariant behaves similarly to the classical invariant on Fadell-Neuwirth fibrations and oriented sphere bundles; on the other hand, we use equivariant homotopy theory to prove that its behavior is wildly different on bundles whose fibers are real projective spaces and whose structure groups are special orthogonal groups. We also explore several other features of our invariant and its relationships with various other invariants motivated by topological robotics.
| Comments: | Minor changes made based on the referee report. To appear in the SIAM Journal on Applied Algebra and Geometry. May differ slightly from the published version |
| Subjects: | Algebraic Topology (math.AT) |
| MSC classes: | 55S40, 55M30, (Primary) 55R10, 70Q05, 55R25, 55R91(Secondary) |
| Cite as: | arXiv:2508.17629 [math.AT] |
| (or arXiv:2508.17629v2 [math.AT] for this version) | |
| https://doi.org/10.48550/arXiv.2508.17629 arXiv-issued DOI via DataCite |
From: Navnath Daundkar [view email]
[v1]
Mon, 25 Aug 2025 03:23:20 UTC (35 KB)
[v2]
Fri, 22 May 2026 07:12:25 UTC (37 KB)
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