Abstract. We study the Z2-equivariant K-theory of M(A), where M(A) is the complement of the complexification of a real hyperplane arrangement, and Z2 acts on M(A) by complex conjugation. We compute the rational equivariant K and KO rings of M(A), and we give two different combinatorial descriptions of a subring Line(A) of the integral equivariant KO ring, where Line(A) is defined to be the subr...

In [3] Felder and Veselov considered the standard and twisted actions of a finite Coxeter group W on the cohomology H(MW ) of the complement of the complexified hyperplane arrangement MW of W . The twisted action is obtained by combining the standard action with complex conjugation; we refer the reader to [3] for precise statements. In a case by case argument, Felder and Veselov obtain a formul...

On a rationally connected variety we would like to use rational curves to obtain a similar result. Complete intersection curves are essentially never rational. (For instance, if X ⊂ P is a hypersurface then a general complete intersection with hyperplanes is rational iff X is a hyperplane or a quadric.) Therefore we have to proceed in a quite different way. Let X be a smooth, projective, ration...

Let ∆ be a dual polar space of rank n ≥ 4, H be a hyperplane of ∆ and Γ := ∆\H be the complement of H in ∆. We shall prove that, if all lines of ∆ have more than 3 points, then Γ is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar spaces, arise from embeddings.

This paper concerns the homotopy type of hyperplane arrangements associated to infinite Coxeter groups acting as reflection groups on Cn. A long-standing conjecture states that the complement of such an arrangement should be aspherical. Some partial results on this conjecture were previously obtained by the author and M. Davis. In this paper, we extend those results to another class of Coxeter ...

In this paper we show that the equivalence of the Apollonian metric and its inner metric remains unchanged by the removal of a point from the domain. For this we need to assume that the complement of the domain is not contained in a hyperplane. This improves a result of the authors wherein the same conclusion was reached under the stronger assumption that the domain contains an exterior point.

Our main result describes how to extend a matroid so that its ground set is a modular hyperplane of the larger matroid. This result yields a new way to view Dowling lattices and new results about line-closed geometries. We complement these topics by showing that line-closure gives simple geometric proofs of the (mostly known) basic results about Dowling lattices. We pursue the topic of line-clo...

Let W be a finite Coxeter group, realized as a group generated by reflections in the /-dimensional Euclidean space V. Let s/ be the hyperplane arrangement in C* = F(g)RC consisting of the complexifications of the reflecting hyperplanes of W in V. The hyperplane complement M = Mw = C l — {JHejl/H has been studied by Arnold [1] in the case when W = Sh Brieskorn [3], Deligne [7] and Orlik and Solo...

For a real hyperplane arrangement A ⊂ R, among the first invariants that were determined for A were the number of chambers in the complement Rn\A by Zavslavsky [Za] and the number of bounded chambers by Crapo [Cr]. In the consideration of certain classes of hypergeometric functions, there also arise arrangements of hypersurfaces which need not be hyperplanes (see e.g. Aomoto [Ao]). In this pape...

Let L be the noncrossing partition lattice associated to a finite Coxeter group W. In this paper we construct explicit bases for top homology groups of intervals and rank-selected subposets L. We define multiplicative structure on Whitney in terms basis, resulting algebra has similarities Orlik-Solomon algebra. As an application, obtain four chain complexes which compute integral Milnor fibre r...