We compute the complete Fadell-Husseini index of the 8 element dihedral group D_8 acting on S^d times S^d, both for F_2 and for integer coefficients. This establishes the complete goup cohomology lower bounds for the two hyperplane case of Grunbaums
1960 mass partition problem: For which d and j can any j arbitrary measures be cut into four equal parts each by two suitably-chosen hyperplanes in R^d? In both cases, we find that the ideal bounds are not stronger than previously established bounds based on one of the maximal abelian subgroups of D_8.
We show that for every injective continuous map f: S^2 --> R^3 there are four distinct points in the image of f such that the convex hull is a tetrahedron with the property that two opposite edges have the same length and the other four edges are als
o of equal length. This result represents a partial result for the topological Borsuk problem for R^3. Our proof of the geometrical claim, via Fadell-Husseini index theory, provides an instance where arguments based on group cohomology with integer coefficients yield results that cannot be accessed using only field coefficients.
It is an amazing and a bit counter-intuitive discovery by Micha Perles from the sixties that there are ``non-rational polytopes: combinatorial types of convex polytopes that cannot be realized with rational vertex coordinates. We describe a simple
construction of non-rational polytopes that does not need duality (Perles ``Gale diagrams): It starts from a non-rational point configuration in the plane, and proceeds with so-called Lawrence extensions. We also show that there are non-rational polyhedral surfaces in 3-space, a discovery by Ulrich Brehm from 1997. His construction also starts from any non-rational point configuration in the plane, and then performs what one should call Brehm extensions, in order to obtain non-rational partial surfaces. These examples and objects are first mile stones on the way to the remarkable universality theorems for polytopes and for polyhedral surfaces by Mnev (1986), Richter-Gebert (1994), and Brehm (1997).
We introduce a deformed product construction for simple polytopes in terms of lower-triangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that spe
cified faces (e.g. all the k-faces) are ``strictly preserved under projection. Thus, starting from an arbitrary neighborly simplicial (d-2)-polytope Q on n-1 vertices we construct a deformed n-cube, whose projection to the last dcoordinates yields a neighborly cubical d-polytope. As an extension of thecubical case, we construct matrix representations of deformed products of(even) polygons (DPPs), which have a projection to d-space that retains the complete (lfloor tfrac{d}{2} rfloor - 1)-skeleton. In both cases the combinatorial structure of the images under projection is completely determined by the neighborly polytope Q: Our analysis provides explicit combinatorial descriptions. This yields a multitude of combinatorially different neighborly cubical polytopes and DPPs. As a special case, we obtain simplified descriptions of the neighborly cubical polytopes of Joswig & Ziegler (2000) as well as of the ``projected deformed products of polygons that were announced by Ziegler (2004), a family of 4-polytopes whose ``fatness gets arbitrarily close to 9.
In 1989 Kalai stated the three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the ``$3^d$-conjecture. It is well-known that the three conjectures
hold in dimensions d leq 3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conjectures B and C fail in all dimensions d geq 5.
