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.
