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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.

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:

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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This theorem is proved in no. 6) below: the method of proof, which is of independent interest, consists in making a parametrix for the exterior derivative by averaging over flows and then employing the theory of compact operators.In no. 1) we define the filtration and in no. 2) by using the theory of exact couples reduce the finiteness problem to an equivalent simpler form (Lemma 3). In no. 3) the relevant results from functional analysis are recalled,