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.
A polytope P of 3-space, which meets a given lattice L only in its vertices, is called L-elementary. An L-elementary tetrahedron has volume ≥ (1/6). det(L), in case equality holds it is called L-primitive. A result of Knudsen, Mumford and Waterman, tells us that any convex polytope P admits a linear simplicial subdivision into tetrahedra which are primitive with respect to the bigger lattice (1/2) t .L, for some t depending on P . Improving this, we show that in fact the lattice (1/4).L always suffices. To this end, we first characterize all L-elementary tetrahedra for which even the intermediate lattice (1/2).L suffices.
Abstract.The «-skeleton of a (2« + 2)-simplex does not embed in R " . This well-known result is due (independently) to van Kampen, 1932, andFlores, 1933, who proved the case p = 2 of the following:
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