Twisted cohomology of arrangements of lines and Milnor fibers

Let \A be an arrangement of affine lines in \C^2, with complement \M(\A). The (co)homo-logy of \M(\A) with twisted coefficients is strictly related to the cohomology of the Milnor fibre associated to the conified arrangement, endowed with the geometric monodromy. Although several partial results are known, even the first Betti number of the Milnor fiber is not understood. We give here a vanishing conjecture for the first homology, which is of a different nature with respect to the known results. Let $\Gamma$ be the graph of \emph{double points} of \A: we conjecture that if $\Gamma$ is connected then the geometric monodromy acts trivially on the first homology of the Milnor fiber (so the first Betti number is combinatorially determined in this case). This conjecture depends only on the combinatorics of \A. We prove it in some cases with stronger hypotheses. In the final parts, we introduce a new description in terms of the group given by the quotient ot the commutator subgroup of \pi_1(\M(\A)) by the commutator of its \emph{length zero subgroup.} We use that to deduce some new interesting cases of a-monodromicity, including a proof of the conjecture under some extra conditions.Comment: 2 m pages, 7 figure

Combinatorial Morse theory and minimality of hyperplane arrangements

We find an explicit combinatorial gradient vector field on the well known complex S (Salvetti complex) which models the complement to an arrangement of complexified hyperplanes. The argument uses a total ordering on the facets of the stratification of R^n associated to the arrangement, which is induced by a generic system of polar coordinates. We give a combinatorial description of the singular facets, finding also an algebraic complex which computes local homology. We also give a precise construction in the case of the braid arrangement.Comment: 29 page

Cohomology of Artin groups of type tilde{A}_n, B_n and applications

We consider two natural embeddings between Artin groups: the group G_{tilde{A}_{n-1}} of type tilde{A}_{n-1} embeds into the group G_{B_n} of type B_n; G_{B_n} in turn embeds into the classical braid group Br_{n+1}:=G_{A_n} of type A_n. The cohomologies of these groups are related, by standard results, in a precise way. By using techniques developed in previous papers, we give precise formulas (sketching the proofs) for the cohomology of G_{B_n} with coefficients over the module Q[q^{+-1},t^{+-1}], where the action is (-q)-multiplication for the standard generators associated to the first n-1 nodes of the Dynkin diagram, while is (-t)-multiplication for the generator associated to the last node. As a corollary we obtain the rational cohomology for G_{tilde{A}_n} as well as the cohomology of Br_{n+1} with coefficients in the (n+1)-dimensional representation obtained by Tong, Yang and Ma. We stress the topological significance, recalling some constructions of explicit finite CW-complexes for orbit spaces of Artin groups. In case of groups of infinite type, we indicate the (few) variations to be done with respect to the finite type case. For affine groups, some of these orbit spaces are known to be K(pi,1) spaces (in particular, for type tilde{A}_n). We point out that the above cohomology of G_{B_n} gives (as a module over the monodromy operator) the rational cohomology of the fibre (analog to a Milnor fibre) of the natural fibration of K(G_{B_n},1) onto the 2-torus.Comment: This is the version published by Geometry & Topology Monographs on 22 February 200

The genus of the configuration spaces for Artin groups of affine type

Let $(W,S)$ be a Coxeter system, $S$ finite, and let $G_{W}$ be the associated Artin group. One has configuration spaces $Y,\ Y_{W},$ where $G_{W}=\pi_1(Y_{W}),$ and a natural $W$-covering $f_{W}:\ Y\to Y_{W}.$ The Schwarz genus $g(f_{W})$ is a natural topological invariant to consider. In this paper we generalize this result by computing the Schwarz genus for a class of Artin groups, which includes the affine-type Artin groups. Let $K=K(W,S)$ be the simplicial scheme of all subsets $J\subset S$ such that the parabolic group $W_J$ is finite. We introduce the class of groups for which $dim(K)$ equals the homological dimension of $K,$ and we show that $g(f_{W})$ is always the maximum possible for such class of groups. For affine Artin groups, such maximum reduces to the rank of the group. In general, it is given by $dim(X_{W})+1,$ where $X_{ W}\subset Y_{ W}$ is a well-known $CW$-complex which has the same homotopy type as $Y_{ W}.$Comment: To appear in Atti Accad. Naz. Lincei Rend. Lincei Mat. App

Cohomology of affine Artin groups and applications

The result of this paper is the determination of the cohomology of Artin groups of type A_n, B_n and \tilde{A}_{n} with non-trivial local coefficients. The main result is an explicit computation of the cohomology of the Artin group of type B_n with coefficients over the module \Q[q^{\pm 1},t^{\pm 1}]. Here the first (n-1) standard generators of the group act by (-q)-multiplication, while the last one acts by (-t)-multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiro's lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type \tilde{A}_{n} as well as the cohomology of the classical braid group {Br}_{n} with coefficients in the n-dimensional representation presented in \cite{tong}. The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be K(\pi,1) spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.Comment: 21 pages, 4 figure

Quantification of errors in large-eddy simulations of a spatially-evolving mixing layer

A stochastic approach based on generalized Polynomial Chaos (gPC) is used to quantify the error in Large-Eddy Simulation (LES) of a spatially-evolving mixing layer flow and its sensitivity to different simulation parameters, viz. the grid stretching in the streamwise and lateral directions and the subgrid scale model constant ($C_S$). The error is evaluated with respect to the results of a highly resolved LES (HRLES) and for different quantities of interest, namely the mean streamwise velocity, the momentum thickness and the shear stress. A typical feature of the considered spatially evolving flow is the progressive transition from a laminar regime, highly dependent on the inlet conditions, to a fully-developed turbulent one. Therefore the computational domain is divided in two different zones (\textit{inlet dependent} and \textit{fully turbulent}) and the gPC error analysis is carried out for these two zones separately. An optimization of the parameters is also carried out for both these zones. For all the considered quantities, the results point out that the error is mainly governed by the value of the $C_S$ constant. At the end of the inlet-dependent zone a strong coupling between the normal stretching ratio and the $C_S$ value is observed. The error sensitivity to the parameter values is significantly larger in the inlet-dependent upstream region; however, low error values can be obtained in this region for all the considered physical quantities by an ad-hoc tuning of the parameters. Conversely, in the turbulent regime the error is globally lower and less sensitive to the parameter variations, but it is more difficult to find a set of parameter values leading to optimal results for all the analyzed physical quantities

A note on the Lawrence-Krammer-Bigelow representation

A very popular problem on braid groups has recently been solved by Bigelow and Krammer, namely, they have found a faithful linear representation for the braid group B_n. In their papers, Bigelow and Krammer suggested that their representation is the monodromy representation of a certain fibration. Our goal in this paper is to understand this monodromy representation using standard tools from the theory of hyperplane arrangements. In particular, we prove that the representation of Bigelow and Krammer is a sub-representation of the monodromy representation which we consider, but that it cannot be the whole representation.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-24.abs.htm

Combinatorial polar orderings and recursively orderable arrangements

Polar orderings arose in recent work of Salvetti and the second author on minimal CW-complexes for complexified hyperplane arrangements. We study the combinatorics of these orderings in the classical framework of oriented matroids, and reach thereby a weakening of the conditions required to actually determine such orderings. A class of arrangements for which the construction of the minimal complex is particularly easy, called {\em recursively orderable} arrangements, can therefore be combinatorially defined. We initiate the study of this class, giving a complete characterization in dimension 2 and proving that every supersolvable complexified arrangement is recursively orderable.Comment: 27 pages, 4 figure