The authors prove that any point configuration whose convex-hull intersection combinatorics is captured by a Radon pair yields a Fan-type covering or labeling theorem for the sphere, and derive colorful, continuous, and (Z/2)^2-equivariant versions.
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abstract
We construct a free $\mathbb{Z}_2$-manifold $X_n$ for a positive integer $n$ such that $w_1(X_n)^n \neq 0$, but there is no $\mathbb{Z}_2$-equivariant map from $S^2$ to $X_n$.
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Covering and labeling generalizations of the Borsuk-Ulam theorem
The authors prove that any point configuration whose convex-hull intersection combinatorics is captured by a Radon pair yields a Fan-type covering or labeling theorem for the sphere, and derive colorful, continuous, and (Z/2)^2-equivariant versions.