Compact Group Actions on Topological and Noncommutative Joins

We consider the Type 1 and Type 2 noncommutative Borsuk-Ulam conjectures of Baum, D$\text{k}a$browski, and Hajac: there are no equivariant morphisms $A \to A \circledast_\delta H$or$H \to A \circledast_\delta H$,respectively,when$H$isanontrivialcompactquantumgroupactingfreelyonaunital$C^*$-algebra$A$\mathrm{.}Here$A \circledast_\delta H$ 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 $A \circledast_\delta H$\mathrm{.}Fortheclassicalcase$H = $\mathcal{C}$(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.