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Genre/Form: | Thèses et écrits académiques |
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Material Type: | Document, Thesis/dissertation, Internet resource |
Document Type: | Internet Resource, Computer File |
All Authors / Contributors: |
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). |
OCLC Number: | 1085239110 |
Notes: | Titre provenant de l'écran-titre. |
Description: | 1 online resource |
Responsibility: | Noémie Combe ; sous la direction de Bernard Coupet. |
Abstract:
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.
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