Equivariant topological complexity

Abstract

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. 55M99, 57S10; 55M30, 55R91

Symmetrized topological complexity

“…We will develop some properties of the equivariant sectional category of a G-fibration, introduced in [5]. These will be applied in later sections to study the symmetrized topological complexity.…”

“…The goal of the current paper is to begin a systematic study of TC Σ (X), by placing it in the framework of equivariant sectional category (as introduced by Colman and Grant in [5]). This allows the use of equivariant obstruction theory to give the following upper bound, entirely analogous to the upper bound for ordinary topological complexity given in [10,Theorem 5.2].…”

Invariant topological complexity

“…Topological complexity now easily translates into the language of the Clapp-Puppe invariant defined in Definition 2.1, as was shown in [11] for the non-equivariant case and not explicitly in [3] for the equivariant one. For clarity of the exposition, we present its proof.…”

“…An answer is not that simple as it may look like and is not unique. We define an invariant, different than the equivariant topological complexity introduced by Colman and Grant in [4], called the invariant topological complexity. By showing its properties we would like to demonstrate that in many situations it better suits into the given frame than that of [4].…”

“…It would be natural to define the equivariant complexity by assuming that all maps considered are G-maps. This approach has been studied in [4]. We will use the notation introduced there and denote the invariant by T C G (X).…”