In this paper we study the Z/2 action on real Grassmann manifolds Gn(R 2n ) and Gn(R 2n ) given by taking (appropriately oriented) orthogonal complement. We completely evaluate the related Z/2 Fadell-Husseini index utilizing a novel computation of the Stiefel-Whitney classes of the wreath product of a vector bundle. These results are used to establish the following geometric result about the orthogonal shadows of a convex body: For n = 2 a (2b + 1), k = 2 a+1 − 1, C a convex body in R 2n , and k real valued functions α 1 , . . . , α k continuous on convex bodies in R 2n with respect to the Hausdorff metric, there exists a subspace V ⊆ R 2n such that projections of C to V and its orthogonal complement V ⊥ have the same value with respect to each function α i , which is α i (p V (C)) = α i (p V ⊥ (C)) for all 1 ≤ i ≤ k.

We prove general results which include classical facts about 60 Pascal's lines as special cases. Along similar lines we establish analogous results about configurations of 2520 conics arising from Mystic Octagon. We offer a more combinatorial outlook on these results and their dual statements. Bézout's theorem is the main tool, however its application is guided by the empirical evidence and computer experiments with program Cinderella. We also emphasize a connection with k-nets of algebraic curves.

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Abstract. Following Ghomi and Tabachnikov's 2008 work, we study the invariant N (M n ) defined as the smallest dimension N such that there exists a totally skew embedding of a smooth manifold M n in R N . This problem is naturally related to the question of estimating the geometric dimension of the stable normal bundle of the configuration space F 2 (M n ) of ordered pairs of distinct points in M n . We demonstrate that in a number of interesting cases the lower bounds on N (M n ) obtained by this method are quite accurate and very close to the best known general upper bound N (M n ) ≤ 4n+1 established by Ghomi and Tabachnikov. We also provide some evidence for the conjecture that for every n-dimensional, compact smooth manifold M n (n > 1),

Let k be the field Fp or the ring Z. We study combinatorial and topological properties of the universal complexes X(k n ) and K(k n ) whose simplices are certain unimodular subsets of k n . We calculate their f -vectors, show that they are shellable but not shifted, and find their applications in toric topology and number theory.Using discrete Morse theory, we detect that X(k n ), K(k n ) and the links of their simplicies are homotopy equivalent to a wedge of spheres specifying the exact number of spheres in the corresponding wedge decompositions. This is a generalisation of Davis and Januszkiewicz's result that K(Z n ) and K(F n 2 ) are (n − 2)-connected simplicial complexes.