Compact group actions on topological and noncommutative joins

We consider the Type 1 and Type 2 noncommutative Borsuk-Ulam conjectures of Baum, Dabrowski, and Hajac: there are no equivariant morphisms A → A ⊛_δ H or H → A ⊛_δ H, respectively, when H is a nontrivial compact quantum group acting freely on a unital C^∗-algebra A. Here A ⊛_δ H denotes the equivariant noncommutative join of A and H; this join procedure is a modification of the topological join that allows a free action of H on A to produce a free action of H on A ⊛_δ H. For the classical case H = C(G), G a compact group, we present a reduction of the Type 1 conjecture and counterexamples to the Type 2 conjecture. We also present some examples and conditions under which the Type 2 conjecture does hold.