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

Abstract

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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.

How to cite

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Napolitano, 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 -

References

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