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

Fabien Napolitano^{[1]}

- [1] Université Paris IX-Dauphine, CEREMADE, UMR CNRS 7534, Place du Maréchal DeLattre de Tassigny, 75776 Paris Cedex 16 (France)

Annales de l’institut Fourier (2003)

- Volume: 53, Issue: 3, page 927-940
- ISSN: 0373-0956

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topNapolitano, Fabien. "Cohomology rings of spaces of generic bipolynomials and extended affine Weyl groups of serie $A$." Annales de l’institut Fourier 53.3 (2003): 927-940. <http://eudml.org/doc/116059>.

@article{Napolitano2003,

abstract = {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.},

affiliation = {Université Paris IX-Dauphine, CEREMADE, UMR CNRS 7534, Place du Maréchal DeLattre de Tassigny, 75776 Paris Cedex 16 (France)},

author = {Napolitano, Fabien},

journal = {Annales de l’institut Fourier},

keywords = {extended affine Weyl groups; bipolynomials; rational functions; stable cohomology rings; extended affine Weyl group; bipolynomial; cohomology ring},

language = {eng},

number = {3},

pages = {927-940},

publisher = {Association des Annales de l'Institut Fourier},

title = {Cohomology rings of spaces of generic bipolynomials and extended affine Weyl groups of serie $A$},

url = {http://eudml.org/doc/116059},

volume = {53},

year = {2003},

}

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

PB - Association des Annales de l'Institut Fourier

VL - 53

IS - 3

SP - 927

EP - 940

AB - 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.

LA - eng

KW - extended affine Weyl groups; bipolynomials; rational functions; stable cohomology rings; extended affine Weyl group; bipolynomial; cohomology ring

UR - http://eudml.org/doc/116059

ER -

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