Tverberg Partitions and Borsuk-ulam Theorems

An N-dimensional real representation E of a finite group G is said to have the “Borsuk-Ulam Property” if any continuous G-map from the (N + 1)-fold join of G (an N-complex equipped with the diagonal G-action) to E has a zero. This happens iff the “Van Kampen characteristic class” of E is nonzero, so using standard computations one can explicitly characterize representations having the B-U property. As an application we obtain the “continuous” Tverberg theorem for all prime powers q, i.e., that some q disjoint faces of a (q − 1)(d + 1)-dimensional simplex must intersect under any continuous map from it into affine d-space. The “classical” Tverberg, which makes the same assertion for all linear maps, but for all q, is explained in our set-up by the fact that any representation E has the analogously defined “linear B-U property” iff it does not contain the trivial representation.