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  E-mail Message: I thought you might be interested in this item at http://www.worldcat.org/oclc/1085239110 Title: On a new cell decomposition of a complement of the discriminant variety : application to the cohomology of braid groups Author: Noémie Combe; Bernard Coupet; Leila Schneps; Athanase Papadopoulos; Erwan Rousseau; Aix-Marseille Université; Ecole Doctorale Mathématiques et Informatique de Marseille (Marseille); Institut de mathématiques de Marseille (I2M) Publisher: 2018. OCLC:1085239110

  
  

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On a new cell decomposition of a complement of the discriminant variety : application to the cohomology of braid groups

Author:	Noémie Combe; Bernard Coupet; Leila Schneps; Athanase Papadopoulos; Erwan Rousseau; All authors
Publisher:	2018.
Dissertation:	Thèse de doctorat : Mathématiques : Aix-Marseille : 2018.
Edition/Format:	Computer file : Document : Thesis/dissertation : English
Summary:	Cette thèse concerne principalement deux objets classiques étroitement liés: d'une part la variété des polynômes complexes unitaires de degré d>1 à une variable, et à racines simples (donc de discriminant différent de zéro), et d'autre part, les groupes de tresses d'Artin avec d brins. Le travail présenté dans cette thèse propose une nouvelle approche permettant des calculs cohomologiques explicites à coefficients dans n'importe quel faisceau. En vue de calculs cohomologiques explicites, il est souhaitable d'avoir à sa disposition un bon recouvrement au sens de Čech. L'un des principaux objectifs de cette thèse est de construire un tel recouvrement basé sur des graphes (appelés signatures) qui rappellent les `dessins d'enfant' et qui sont associées aux polynômes complexes classifiés par l'espace de polynômes. Cette décomposition de l'espace de polynômes fournit une stratification semi-algébrique. Le nombre de composantes connexes de chaque strate est calculé dans le dernier chapitre ce cette thèse. Néanmoins, cette partition ne fournit pas immédiatement un recouvrement adapté au calcul de la cohomologie de Čech (avec n'importe quels coefficients) pour deux raisons liées et évidentes: d'une part les sous-ensembles du recouvrement ne sont pas ouverts, et de plus ils sont disjoints puisqu'ils correspondent à différentes signatures. Ainsi, l'objectif principal du chapitre 6 est de "corriger'' le recouvrement de départ afin de le transformer en un bon recouvrement ouvert, adapté au calcul de la cohomologie Čech. Cette construction permet ensuite un calcul explicite des groupes de cohomologie de Čech à valeurs dans un faisceau localement constant. This thesis mainly concerns two closely related classical objects: on the one hand, the variety of unitary complex polynomials of degree d>1 with a variable, and with simple roots (hence with a non-zero discriminant), and on the other hand, the d strand Artin braid groups. The work presented in this thesis proposes a new approach allowing explicit cohomological calculations with coefficients in any sheaf. In order to obtain explicit cohomological calculations, it is necessary to have a good cover in the sense of Čech. One of the main objectives of this thesis is to construct such a good covering, based on graphs that are reminiscent of the ''dessins d'enfants'' and which are associated to the complex polynomials. This decomposition of the space of polynomials provides a semi-algebraic stratification. The number of connected components in each stratum is counted in the last chapter of this thesis. Nevertheless, this partition does not immediately provide a ''good'' cover adapted to the computation of the cohomology of Čech (with any coefficients) for two related and obvious reasons: on the one hand the subsets of the cover are not open, and moreover they are disjoint since they correspond to different signatures. Therefore, the main purpose of Chapter 6 is to ''correct'' the cover in order to transform it into a good open cover, suitable for the calculation of the Čech cohomology. It is explicitly verified that there is an open cover such that all the multiple intersections are contractible. This allows an explicit calculation of cohomology groups of Čech with values in a locally constant sheaf.  Read more...
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Subjects	Cohomologie. Polynômes. Tresses, Théorie des. View all subjects
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