Cohomology rings of spaces of generic bipolynomials and extended affine Weyl groups of serie $A$

Napolitano, Fabien

Annales de l'Institut Fourier, Tome 53 (2003) no. 3, pp. 927-940.

Un bipolynôme est une application holomorphe d’une sphère dans une sphère telle qu’un des points de la sphère image ait exactement deux préimages. Les invariants topologiques de l’espace des bipolynômes sans racines multiples sont reliés aux classes caractéristiques des fonctions rationnelles avec deux pôles et aux groupes de tresses généralisés associés aux extensions des groupes de Weyl affines de la série $A$. Nous prouvons que les anneaux de cohomologie de l’espace des bipolynômes de bidegré $(k,l)$ se stabilisent quand $k$ tend vers l’infini et que les anneaux stables correspondant à différents $l$ se stabilisent quand $l$ tend vers l’infini. De plus nous prouvons un analogue de la décomposition de Snaith pour les groupes de cohomologie stables. Les deux premiers termes de la suite d’anneaux de cohomologie stable sont les mêmes que pour les singularités simples de types $A$ et $B$. Les autres termes sont encore inconnus.

A bipolynomial is a holomorphic mapping of a sphere onto a sphere such that some point on the target sphere has exactly two preimages. The topological invariants of spaces of bipolynomials without multiple roots are connected with characteristic classes of rational functions with two poles and generalized braid groups associated to extended affine Weyl groups of the serie $A$. We prove that the cohomology rings of the spaces of bipolynomials of bidegree $(k,l)$ stabilize as $k$ tends to infinity and that the stable cohomology rings obtained for different $l$ also stabilize as $l$ tends to infinity. Moreover we prove an analog of Snaith splitting formula for the stable cohomology groups. The first terms of the sequence of stable cohomology rings are the same as the stable cohomology rings of the simple singularities of types $A$ and $B$. Other terms of the sequence are still unknown.

Classification : 55R80, 55R40, 20F36, 20F55
Keywords: extended affine Weyl groups, bipolynomials, rational functions, stable cohomology rings
Mot clés : groupes de Weyl affines étendus, bipolynômes, fonctions rationnelles, anneaux de cohomologie stable

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