We define and study an equivariant version of Farber’s topological complexity for spaces with a given compact group action. This is a special case of the equivariant sectional category of an equivariant map, also defined in this paper. The relationship of these invariants with the equivariant Lusternik–Schnirelmann category is given. Several examples and computations serve to highlight the similarities and differences with the nonequivariant case. We also indicate how the equivariant topological complexity can be used to give estimates of the nonequivariant topological complexity.
Hellen Colman. Mark Grant. "Equivariant topological complexity." Algebr. Geom. Topol. 12 (4) 2299 - 2316, 2012. https://doi.org/10.2140/agt.2012.12.2299