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PreprintCC BY 4.0arXiv

Parametrized LS category and group actions

  • Urban Ogrinec
  • Petar Pavešić
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Abstract

We introduce a new invariant associated to an action of a group G on a (possibly non-commutative) group K. The invariant is based on the G-parametrized LS-category of the semi-direct product Krtimes G but it also depends on a choice of a derivation d: G→ K. The invariant is thus denoted tc(d) and it takes integer values between cd(K) and cd(K)+cd(G) (where cd(G) stands for the cohomological dimension of a group). We develop a general theory and then compute its values for a number of important examples, including fundamental groups of mapping tori, of pure braid groups and of iterated semi-direct products of free groups. To achieve this, we relate tc(d) to the cohomological dimension of its factors, to the sectional category of the inclusion of G in Krtimes G, and to the properties of torsors that are associated to the derivation d.

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Cite

BibTeXRIS
BibTeX
@misc{ogrinec2026parametrized,
  title = {{Parametrized LS category and group actions}},
  author = {Urban Ogrinec and Petar Pavešić},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01462},
  eprint = {2609.01462},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01462},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Parametrized LS category and group actions
AU  - Urban Ogrinec
AU  - Petar Pavešić
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01462
UR  - https://arxiv.org/abs/2609.01462
AB  - We introduce a new invariant associated to an action of a group G on a (possibly non-commutative) group K. The invariant is based on the G-parametrized LS-category of the semi-direct product Krtimes G but it also depends on a choice of a derivation d: G→ K. The invariant is thus denoted tc(d) and it takes integer values between cd(K) and cd(K)+cd(G) (where cd(G) stands for the cohomological dimension of a group). We develop a general theory and then compute its values for a number of important examples, including fundamental groups of mapping tori, of pure braid groups and of iterated semi-direct products of free groups. To achieve this, we relate tc(d) to the cohomological dimension of its factors, to the sectional category of the inclusion of G in Krtimes G, and to the properties of torsors that are associated to the derivation d.
ER  -