Invariants in noncommutative dynamics

When a compact quantum group H coacts freely on unital C^⁎-algebras A and B, the existence of equivariant maps A→B may often be ruled out due to the incompatibility of some invariant. We examine the limitations of using invariants, both concretely and abstractly, to resolve the noncommutative Borsuk–Ulam conjectures of Baum–Dąbrowski–Hajac. Among our results, we find that for certain finite-dimensional H, there can be no well-behaved invariant which solves the Type 1 conjecture for all free coactions of H. This claim is in stark contrast to the case when H is finite-dimensional and abelian. In the same vein, it is possible for all iterated joins of H to be cleft as comodules over the Hopf algebra associated to H. Finally, two commonly used invariants, the local-triviality dimension and the spectral count, may both change in a θ-deformation procedure.