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PreprintCC BY 4.0arXiv

Motion planning invariants and families of subgroups

  • Ekansh Jauhari
  • Ben Knudsen
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Abstract

We introduce a notion of topological complexity of a group with respect to a family of subgroups, with the case of the trivial and diagonal families recovering the classical category and topological complexity, respectively. These invariants come equipped with a battery of upper and lower bounds derived from functoriality and Bredon cohomology. We introduce a new family of subgroups, the permutational family, and show that the corresponding invariant is a lower bound for the distributional topological complexity of Dranishnikov-Jauhari and Knudsen-Weinberger. As a first application, we show that the permutational and diagonal families coincide, and thus that classical and distributional topological complexity are equal, for a large class of torsion-free groups, extending work of Dranishnikov. Second, we show that the lower bounds of Grant-Lupton-Oprea are in fact lower bounds on distributional topological complexity; in particular, it follows that Farber's conjecture holds distributionally.

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Cite

BibTeXRIS
BibTeX
@misc{jauhari2026motion,
  title = {{Motion planning invariants and families of subgroups}},
  author = {Ekansh Jauhari and Ben Knudsen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30579},
  eprint = {2609.30579},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30579},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Motion planning invariants and families of subgroups
AU  - Ekansh Jauhari
AU  - Ben Knudsen
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.30579
UR  - https://arxiv.org/abs/2609.30579
AB  - We introduce a notion of topological complexity of a group with respect to a family of subgroups, with the case of the trivial and diagonal families recovering the classical category and topological complexity, respectively. These invariants come equipped with a battery of upper and lower bounds derived from functoriality and Bredon cohomology. We introduce a new family of subgroups, the permutational family, and show that the corresponding invariant is a lower bound for the distributional topological complexity of Dranishnikov-Jauhari and Knudsen-Weinberger. As a first application, we show that the permutational and diagonal families coincide, and thus that classical and distributional topological complexity are equal, for a large class of torsion-free groups, extending work of Dranishnikov. Second, we show that the lower bounds of Grant-Lupton-Oprea are in fact lower bounds on distributional topological complexity; in particular, it follows that Farber's conjecture holds distributionally.
ER  -