Equivariant Relative Sectional Category and Induced Invariants
Abstract
The relative sectional category, introduced by González, Grant, and Vandembroucq for fibrations and later extended by García-Calcines to arbitrary maps, provides a common framework encompassing several numerical homotopy invariants, including the Lusternik--Schnirelmann category, the topological complexity of a map, and homotopic distance. In this paper, we introduce and study the equivariant analogue of the relative sectional category for G-maps. We establish its fundamental homotopy-theoretic properties, including comparison, product, and composition inequalities, as well as its behavior under changes of domain and codomain. As applications, we introduce and investigate equivariant analogues of the topological complexity of a map, in the sense of Scott and Murillo--Wu, and the equivariant Lusternik--Schnirelmann category of a map. Several examples are provided to illustrate the theory and demonstrate that these invariants extend the corresponding classical equivariant notions.
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BibTeX
@misc{daundkar2026equivariant,
title = {{Equivariant Relative Sectional Category and Induced Invariants}},
author = {Navnath Daundkar and Rekha Santhanam and Bittu Singh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.13663},
eprint = {2608.13663},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.13663},
}
RIS
TY - UNPB
TI - Equivariant Relative Sectional Category and Induced Invariants
AU - Navnath Daundkar
AU - Rekha Santhanam
AU - Bittu Singh
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2608.13663
UR - https://arxiv.org/abs/2608.13663
AB - The relative sectional category, introduced by González, Grant, and Vandembroucq for fibrations and later extended by García-Calcines to arbitrary maps, provides a common framework encompassing several numerical homotopy invariants, including the Lusternik--Schnirelmann category, the topological complexity of a map, and homotopic distance. In this paper, we introduce and study the equivariant analogue of the relative sectional category for G-maps. We establish its fundamental homotopy-theoretic properties, including comparison, product, and composition inequalities, as well as its behavior under changes of domain and codomain. As applications, we introduce and investigate equivariant analogues of the topological complexity of a map, in the sense of Scott and Murillo--Wu, and the equivariant Lusternik--Schnirelmann category of a map. Several examples are provided to illustrate the theory and demonstrate that these invariants extend the corresponding classical equivariant notions.
ER -