The Lusternik-Schnirelman category (L-S category) and the topological complexity are well-known and much-studied numerical invariants of the homotopy type of a topological space. L-S category has a long history beginning with its involvement in questions of critical points of a function on a smooth manifold. Topological complexity is a much more recent notion, with provenance in topological robotics. Despite their different timelines, both invariants may be viewed as specializations of a more general notion in several ways. One such way is to use the notion of \({\mathcal A}\)-category introduced by M. Clapp and D. Puppe [Trans. Am. Math. Soc. 298, 603–620 (1986; Zbl 0618.55003)].
Equivariant notions of both invariants have been developed and studied. Like their non-equivariant counterparts, equivariant L-S category and topological complexity may be viewed as specializations of a more general equivariant \({\mathcal A}\)-category, denoted by \(_{\mathcal A} \mathrm{cat}_G(X)\) for \(X\) a \(G\)-space and \({\mathcal A}\) a class of \(G\)-equivariant subsets of \(X\). If \({\mathcal A}\) is the class of \(G\)-orbits of the action, then \(_{\mathcal A} \mathrm{cat}_G(X)\) corresponds to the equivariant L-S category of \(X\). With care, choosing \({\mathcal A}\) to be the union of all \((G \times G)\)-orbits that meet \(\Delta(X)\), the image of the diagonal map in \(X \times X\), then \(_{\mathcal A} \mathrm{cat}_{G\times G}(X\times X)\) corresponds to the invariant (equivariant) topological complexity of \(X\). (This supplants an earlier notion of equivariant topological complexity.)
The translation groupoid \(G\ltimes X\) is used here (rather than the quotient space of the action) to accommodate non-free actions. An essential equivalence of actions is a map \(\phi\times \epsilon \colon G\ltimes X \to H\ltimes Y\) that preserves certain structures related to the actions. Then two actions are Morita equivalent if there is a zig-zag of essential equivalences between them. Roughly speaking, this means the actions define the same quotient object. Given an equivariant map \(\phi\ltimes \epsilon \colon G\ltimes X \to H\ltimes Y\) with \(\mathcal{A}\) a class of \(G\)-invariant subsets of \(X\), write \(\mathcal{A}'\) for the class of \(H\)-invariant subsets of \(Y\) that consists of the saturations of each set \(\epsilon(A) \subseteq Y\) with respect to the \(H\)-action on \(Y\) for all \(A \in \mathcal{A}\).
The main result is given in Theorem 5.1: Let \(\phi\times \epsilon \colon G\ltimes X \to H\ltimes Y\) be an essential equivalence and \(\mathcal{A}\) a class of \(G\)-invariant subsets of \(X\). Then \(_{\mathcal A} \mathrm{cat}_G(X) = _{\mathcal A'} \mathrm{cat}_H(Y)\). The main aim of the paper – indicated in its title – is then achieved in two corollaries: The equivariant category (Corollary 5.9) and the invariant topological complexity (Corollary 5.10) are both invariant under Morita equivalence for compact Lie group actions on metrizable spaces.