No abstract

Abstract. The ring structure of the integral cohomology of complements of real linear subspace arrangements is considered. While the additive structure of the cohomology is given in terms of the intersection poset and dimension function by a theorem of Goresky and MacPherson, we describe the multiplicative structure in terms of the intersection poset, the dimension function and orientations of the participating subspaces for the class of arrangements without pairs of intersections of codimension one. In particular, this yields a description of the integral cohomology ring of complex arrangements conjectured by Yuzvinsky. For general real arrangements a weaker result is obtained. The approach is geometric and the methods are elementary.

It is shown that the neighborhood complexes of a family of vertex critical subgraphs of Kneser graphs -the stable Kneser graphs introduced by L. Schrijver -are spheres up to homotopy. Furthermore, it is shown that the neighborhood complexes of a subclass of the stable Kneser graphs contain the boundaries of associahedra (simplicial complexes encoding triangulations of a polygon) as a strong deformation retract.

Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master's level and beyond. The books, often well class-tested by their author, may have an informal, personal, even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, into very polished texts.

The well-known "splitting necklace theorem" of Alon [N. Alon, Splitting necklaces, Adv. Math. 63 (1987) 247-253] says that each necklace with k · a i beads of color i = 1, . . . , n, can be fairly divided between k thieves by at most n(k − 1) cuts. Alon deduced this result from the fact that such a division is possible also in the case of a continuous necklace [0, 1] where beads of given color are interpreted as measurable sets A i ⊂ [0, 1] (or more generally as continuous measures μ i ). We demonstrate that Alon's result is a special case of a multidimensional consensus division theorem about n continuous probability measures μ 1 , . . . , μ n on a d-cube [0, 1] d . The dissection is performed by m 1 + · · · + m d = n(k − 1) hyperplanes parallel to the sides of [0, 1] d dividing the cube into m 1 · · · · · m d elementary cuboids (parallelepipeds) where the integers m i are prescribed in advance.