Nullhomotopic equivariant shifts and Borsuk-Ulam-compatible compact abelian groups
We disprove the classical version of the Borsuk-Ulam-flavored conjecture, due to Baum-D{ą}browski-Hajac, that free actions $X\times \mathbb{G}\to X$ on non-empty compact Hausdorff spaces by non-trivial compact groups admit no equivariant maps $X*\mathbb{G}\to X$. Specifically, among non-trivial compact abelian $\mathbb{G}$, the conjecture fails precisely when $\mathbb{G}$ is torsion-free profinite.
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