Equivariant Relative Sectional Category and Induced Invariants

Abstract

The relative sectional category, introduced by Gonz\'alez, Grant, and Vandembroucq for fibrations and later extended by Garc\'ia-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.