Cohomology rings of spaces of generic bipolynomials and extended affine Weyl groups of serie A
[Anneaux de cohomologie des espaces de bipolynômes génériques et groupes de Weyl affines étendus de la série A]
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.

DOI : 10.5802/aif.1966
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
Napolitano, Fabien 1

1 Université Paris IX-Dauphine, CEREMADE, UMR CNRS 7534, Place du Maréchal DeLattre de Tassigny, 75776 Paris Cedex 16 (France)
@article{AIF_2003__53_3_927_0,
     author = {Napolitano, Fabien},
     title = {Cohomology rings of spaces of generic bipolynomials and extended affine {Weyl} groups of serie $A$},
     journal = {Annales de l'Institut Fourier},
     pages = {927--940},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {3},
     year = {2003},
     doi = {10.5802/aif.1966},
     zbl = {1030.55012},
     mrnumber = {2008447},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1966/}
}
TY  - JOUR
AU  - Napolitano, Fabien
TI  - Cohomology rings of spaces of generic bipolynomials and extended affine Weyl groups of serie $A$
JO  - Annales de l'Institut Fourier
PY  - 2003
SP  - 927
EP  - 940
VL  - 53
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1966/
DO  - 10.5802/aif.1966
LA  - en
ID  - AIF_2003__53_3_927_0
ER  - 
%0 Journal Article
%A Napolitano, Fabien
%T Cohomology rings of spaces of generic bipolynomials and extended affine Weyl groups of serie $A$
%J Annales de l'Institut Fourier
%D 2003
%P 927-940
%V 53
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1966/
%R 10.5802/aif.1966
%G en
%F AIF_2003__53_3_927_0
Napolitano, Fabien. Cohomology rings of spaces of generic bipolynomials and extended affine Weyl groups of serie $A$. Annales de l'Institut Fourier, Tome 53 (2003) no. 3, pp. 927-940. doi : 10.5802/aif.1966. https://aif.centre-mersenne.org/articles/10.5802/aif.1966/

[Arn70a] V.I. Arnold On some topological invariants of algebraic functions, Trans. Moscow Math. Soc., Volume 21 (1970), pp. 30-52 | MR | Zbl

[Arn70b] V.I. Arnold Topological invariants of algebraic functions II, Funct. Anal. Appl., Volume 2 (1970), pp. 91-98 | DOI | MR | Zbl

[Arn78] V.I. Arnold Critical points of functions on a manifold with boundary, the simple Lie groups B k , C k and F 4 and singularities of evolutes, Russian Math. Surveys, Volume 33 (1978) no. 5, pp. 99-116 | DOI | MR | Zbl

[Arn96] V.I. Arnold Topological Classification of Trigonometric Polynomials and Combinatorics of Graphs with an Equal Number of Vertices and Edges, Funct. Anal. Appl., Volume 30 (1996) no. 1, pp. 1-14 | DOI | MR | Zbl

[Bri72] E. Brieskorn Sur les groupes de tresses (d'après Arnol'd), Séminaire Bourbaki, Volume 401 (1971/72), pp. 21-44 | Numdam | Zbl

[DZ98] B. Dubrovin; Y. Zhang Extended affine Weyl groups and Frobenius manifolds, Comp. Math., Volume 2 (1998), pp. 167-219 | DOI | MR | Zbl

[Fuc74] D.B. Fuchs Quillenization and bordisms, Funct. Anal. Appl., Volume 8 (1974) no. 1, pp. 31-36 | DOI | Zbl

[Gor78] V.V. Goryunov Cohomology of braid groups of series C and D and certain stratifications, Funct. Anal. Appl., Volume 12 (1978) no. 2, pp. 76-77 | MR | Zbl

[Gor81] V.V. Goryunov Geometry of the bifurcation diagrams of simple projections onto a line, Funct. Anal. Appl., Volume 15 (1981) no. 2, pp. 77-82 | DOI | MR | Zbl

[Gor82] V.V. Goryunov Cohomology of the braid groups of the series C and D, Trans. Moscow Math. Soc., Volume 42 (1982), pp. 232-241 | Zbl

[Knö82] H. Knörrer Zum K(π,1) problem für isolierte singularitäten von vollstandiger durschnitten, Comp. Math., Volume 45 (1982) no. 3, pp. 330-340 | EuDML | Numdam | MR | Zbl

[Lin72] V.Ya. Lin Superpositions of algebraic functions, Funct. Anal. Appl., Volume 6 (1972) no. 3, pp. 240-241 | DOI | MR | Zbl

[Lin76] V.Ya. Lin Superpositions of algebraic functions, Funct. Anal. Appl., Volume 10 (1976) no. 1, pp. 32-38 | DOI | MR | Zbl

[May72] J.P. May The geometry of iterated loop spaces, Lecture Notes in Math., 268, Springer-Verlag, Berlin-New York, 1972 | MR | Zbl

[Nap98] F. Napolitano Discriminant and bifurcation diagram of complex trigonometric polynomials, C.R. Acad. Sci. Paris, Série I, Volume 327 (1998), pp. 771-776 | MR | Zbl

[Seg73] G.B. Segal Configuration spaces and iterated loop-spaces, Invent. Math., Volume 21 (1973), pp. 213-221 | DOI | EuDML | MR | Zbl

[Vas89] V.A. Vassiliev Topological order complexes and resolutions of discriminant sets, Funct. Anal. Appl., Volume 23 (1989), pp. 24-36 | Zbl

[Vas92a] V.A. Vassiliev Complements of Discriminants of Smooth Maps: Topology and Applications, Mathematical Monographs, 98, Amer. Math. Soc., 1992 | MR | Zbl

[Vas92b] V.A. Vassiliev Developments in Mathematics: the Moscow School, chapter Invariants of knots and complements of discriminants (1992), pp. 195-250 | Zbl

Cité par Sources :