50 research outputs found

    The integer cohomology of toric Weyl arrangements

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    A referee found an error in the proof of the Theorem 2 that we could not fix. More precisely, the proof of Lemma 2.1 is incorrect. Hence the fact that integer cohomology of complement of toric Weyl arrangements is torsion free is still a conjecture. ----- A toric arrangement is a finite set of hypersurfaces in a complex torus, every hypersurface being the kernel of a character. In the present paper we prove that if \Cal T_{\wdt W} is the toric arrangement defined by the \textit{cocharacters} lattice of a Weyl group \wdt W, then the integer cohomology of its complement is torsion free

    Blocking Sets in the complement of hyperplane arrangements in projective space

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    It is well know that the theory of minimal blocking sets is studied by several author. Another theory which is also studied by a large number of researchers is the theory of hyperplane arrangements. We can remark that the affine space AG(n,q)AG(n,q) is the complement of the line at infinity in PG(n,q)PG(n,q). Then AG(n,q)AG(n,q) can be regarded as the complement of an hyperplane arrangement in PG(n,q)PG(n,q)! Therefore the study of blocking sets in the affine space AG(n,q)AG(n,q) is simply the study of blocking sets in the complement of a finite arrangement in PG(n,q)PG(n,q). In this paper the author generalizes this remark starting to study the problem of existence of blocking sets in the complement of a given hyperplane arrangement in PG(n,q)PG(n,q). As an example she solves the problem for the case of braid arrangement. Moreover she poses significant questions on this new and interesting problem

    The integer cohomology of toric Weyl arrangements

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    A toric arrangement is a finite set of hypersurfaces in a complex torus, every hypersurface being the kernel of a character. In the present paper we prove that if T(W) is the toric arrangement defined by the cocharacters lattice of a Weyl group W, then the integer cohomology of its complement is torsion free.Arrangement of hyperplanes, toric arrangements, CW complexes, Salvetti complex, Weyl groups, integer cohomology

    A stability-like theorem for cohomology of pure braid groups of the series A, B and D

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    Consider the ring R:=\Q[\tau,\tau^{-1}] of Laurent polynomials in the variable τ\tau. The Artin's Pure Braid Groups (or Generalized Pure Braid Groups) act over R,R, where the action of every standard generator is the multiplication by τ\tau. In this paper we consider the cohomology of such groups with coefficients in the module RR (it is well known that such cohomology is strictly related to the untwisted integral cohomology of the Milnor fibration naturally associated to the reflection arrangement). We give a sort of \textit{stability} theorem for the cohomologies of the infinite series AA, BB and D,D, finding that these cohomologies stabilize, with respect to the natural inclusion, at some number of copies of the trivial RR-module \Q. We also give a formula which compute this number of copies.Comment: 17 pages; added reference for section

    Social choice among complex objects: Mathematical tools

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    Here the reader can find some basic definitions and notations in order to better understand the model for social choise described by L. Marengo and S. Settepanella in their paper: Social choice among complex objects. The interested reader can refer to [Bou68], [Massey] and [OT92] to go into more depth.Arrangements, simplicial complexes, CW complexes,fundamental group, Salvetti's complex.

    The homotopy type of toric arrangements

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    A toric arrangement is a finite set of hypersurfaces in a complex torus, every hypersurface being the kernel of a character. In the present paper we build a CW-complex homotopy equivalent to the arrangement complement, with a combinatorial description similar to that of the well-known Salvetti complex. If the toric arrangement is defined by a Weyl group we also provide an algebraic description, very handy for cohomology computations. In the last part we give a description in terms of tableaux for a toric arrangement appearing in robotics.Comment: To appear on J. of Pure and Appl. Algebra. 16 pages, 3 picture

    Homology graph of real arrangements and monodromy of Milnor Fiber

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    We study the first homology group of the Milnor fiber of sharp arrangements in the real projective plane. Our work relies on the minimal Salvetti complex of the deconing arrangement and its boundary map. We describe an algorithm which computes possible eigenvalues of the first monodromy operator. We prove that, if a condition on some intersection points of lines is satisfied, then the only possible non trivial eigenvalues are cubic roots of the unity. Moreover we give sufficient conditions for just eigenvalues of order 3 or 4 to appear in cases in which this condition is not satisfied.Comment: 35 pages, 15 figure

    Braid groups in complex Grassmannians

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    We describe the fundamental group and second homotopy group of ordered k−k-point sets in Gr(k,n)Gr(k,n) generating a subspace of fixed dimension.Comment: 10 page
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