We present a proof for certain cases of the noncommutative Borsuk-Ulam conjectures proposed by Baum, D\k{ a}browski, and Hajac. When a unital $C^*$-algebra $A$ admits a free action of $\mathbb{ Z}/k\mathbb{ Z}$, $k \geq 2$, there is no equivariant map from $A$ to the $C^*$-algebraic join of $A$ and the compact "quantum" group $C(\mathbb{ Z}/k\mathbb{ Z})$. This also resolves D\k{ a}browski's conjecture on unreduced suspensions of $C^*$-algebras. Finally, we formulate a different type of noncommutative join than the previous authors, which leads to additional open problems for finite cyclic group actions.
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