Salvetti complex, spectral sequences and cohomology of Artin groups
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 2, pp. 267-296.

Le but de ce travail est de donner une brève introduction aux complexes de Salvetti comme instrument pour étudier la cohomologie des groupes d’Artin. Nous montrons comment une suite spectrale donnée par une filtration sur le complexe va définir une méthode, utile ainsi que très naturelle, pour étudier récursivement la cohomologie des groupes d’Artin, avec une grande simplification dans les calculs. Dans la dernière partie du travail nous présentons des exemples d’applications.

The aim of this short survey is to give a quick introduction to the Salvetti complex as a tool for the study of the cohomology of Artin groups. In particular we show how a spectral sequence induced by a filtration on the complex provides a very natural and useful method to study recursively the cohomology of Artin groups, simplifying many computations. In the last section some examples of applications are presented.

@article{AFST_2014_6_23_2_267_0,
     author = {Callegaro, Filippo},
     title = {Salvetti complex, spectral sequences  and cohomology of {Artin} groups},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {267--296},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 23},
     number = {2},
     year = {2014},
     doi = {10.5802/afst.1407},
     zbl = {06297893},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/afst.1407/}
}
TY  - JOUR
AU  - Callegaro, Filippo
TI  - Salvetti complex, spectral sequences  and cohomology of Artin groups
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2014
SP  - 267
EP  - 296
VL  - 23
IS  - 2
PB  - Université Paul Sabatier, Institut de mathématiques
PP  - Toulouse
UR  - http://www.numdam.org/articles/10.5802/afst.1407/
DO  - 10.5802/afst.1407
LA  - en
ID  - AFST_2014_6_23_2_267_0
ER  - 
%0 Journal Article
%A Callegaro, Filippo
%T Salvetti complex, spectral sequences  and cohomology of Artin groups
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2014
%P 267-296
%V 23
%N 2
%I Université Paul Sabatier, Institut de mathématiques
%C Toulouse
%U http://www.numdam.org/articles/10.5802/afst.1407/
%R 10.5802/afst.1407
%G en
%F AFST_2014_6_23_2_267_0
Callegaro, Filippo. Salvetti complex, spectral sequences  and cohomology of Artin groups. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 2, pp. 267-296. doi : 10.5802/afst.1407. http://www.numdam.org/articles/10.5802/afst.1407/

[1] Artin (E.).— Theorie des zöpfe, Abh. Math. Sem. Univ. Hamburg 4, p. 47-72 (1925). | MR

[2] Brieskorn (E.).— Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe, Invent. Math. 12, p. 57-61 (1971). | MR | Zbl

[3] Brieskorn (E.).— Sur les groupes de tresses [d’après V. I. Arnol’d], Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Springer, Berlin, (1973), p. 21-44. Lecture Notes in Math., Vol. 317. | Numdam | MR | Zbl

[4] Brown (K. S.).— Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994, Corrected reprint of the 1982 original. | MR | Zbl

[5] Björner (A.), Ziegler (G. M.).— Combinatorial stratification of complex arrangements, J. Amer. Math. Soc. 5, no. 1, p. 105-149 (1992). | MR | Zbl

[6] Callegaro (F.).— On the cohomology of Artin groups in local systems and the associated Milnor fiber, J. Pure Appl. Algebra 197, no. 1-3, p. 323-332 (2005). | MR | Zbl

[7] Callegaro (F.).— The homology of the Milnor fiber for classical braid groups, Algebr. Geom. Topol. 6, p. 1903-1923 (electronic) (2006). | MR | Zbl

[8] Callegaro (F.), Cohen (F.), Salvetti (M.).— The cohomology of the braid group B 3 and of SL 2 () with coefficients in a geometric representation, Quart. J. Math. 64, p. 847-889 (2013). | MR

[9] Charney (R.), Davis (M. W.).— The K(π,1)-problem for hyperplane complements associated to infinite reflection groups, J. Amer. Math. Soc. 8, no. 3, p. 597-627 (1995). | MR | Zbl

[10] Cohen (D.), Denham (G.), Falk (M.), Suciu (A. I.), Terao (H.), Yuzvinsky (S.).— Complex Arrangements: Algebra, Geometry, Topology, 2009 (work in progress), available at http://www.math.uiuc.edu/~schenck/cxarr.pdf.

[11] Callegaro (F.), Moroni (D.), Salvetti (M.).— Cohomology of affine Artin groups and applications, Trans. Amer. Math. Soc. 360, no. 8, p. 4169-4188 (2008). | MR | Zbl

[12] Callegaro (F.), Moroni (D.), Salvetti (M.).— The K(π,1) problem for the affine Artin group of type B ˜ n and its cohomology, J. Eur. Math. Soc. (JEMS) 12, no. 1, p. 1-22 (2010). | MR | Zbl

[13] De Concini (C.), C. Procesi (C.), M. Salvetti (M.).— Arithmetic properties of the cohomology of braid groups, Topology 40, no. 4, p. 739-751 (2001). | MR | Zbl

[14] De Concini (C.), Salvetti (M.).— Cohomology of Artin groups: Addendum: “The homotopy type of Artin groups" [Math. Res. Lett. 1, no. 5, p. 565-577 (1994)] by Salvetti, Math. Res. Lett. 3, no. 2, p. 293-297 (1996). | MR | Zbl

[15] De Concini (C.), Salvetti (M.), Stumbo (F.).— The top-cohomology of Artin groups with coefficients in rank-1 local systems over , Topology Appl. 78, no. 1-2, p. 5-20 (1997), Special issue on braid groups and related topics (Jerusalem, 1995). | MR | Zbl

[16] Deligne (P.).— Les immeubles des groupes de tresses généralisés, Invent. Math. 17, p. 273-302 (1972). | MR | Zbl

[17] Dickson (L. E.).— A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. Amer. Math. Soc. 12, no. 1, p. 75-98 (1911). | MR

[18] Fox (R.), L. Neuwirth (L.).— The braid groups, Math. Scand. 10, p. 119-126 (1962). | MR | Zbl

[19] Frenkel’ (È. V.).— Cohomology of the commutator subgroup of the braid group, Funktsional. Anal. i Prilozhen. 22, no. 3, p. 91-92 (1988). | MR | Zbl

[20] Fuks (D. B.).— Cohomology of the braid group mod 2, Funct. Anal. Appl. 4, no. 2, p. 143-151 (1970). | MR | Zbl

[21] Gel’fand (I. M.), Rybnikov (G. L.).— Algebraic and topological invariants of oriented matroids, Dokl. Akad. Nauk SSSR 307, no. 4, p. 791-795 (1989). | MR | Zbl

[22] Humphreys (J. E.).— Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge (1990). | MR | Zbl

[23] Markaryan (N. S.).— Homology of braid groups with nontrivial coefficients, Mat. Zametki 59, no. 6, p. 846-854, 960 (1996). | MR | Zbl

[24] Matsumoto (H.).— Générateurs et relations des groupes de Weyl généralisés, C. R. Acad. Sci. Paris 258, p. 3419-3422 (1964). | MR | Zbl

[25] Milnor (J.).— Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J. (1968). | MR | Zbl

[26] Milnor (J.).— Introduction to algebraic K-theory, Princeton University Press, Princeton, N.J., Annals of Mathematics Studies, No. 72 (1971). | MR | Zbl

[27] Magnus (W.), Abraham Karrass (A.), Solitar (D.).— Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney (1966). | MR | Zbl

[28] Orlik (P.), Terao (H.).— Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin (1992). | MR | Zbl

[29] Paris (L.).— K(π,1) conjecture for Artin groups, Proceedings of the conference “Arrangements in Pyrénées" held in Pau (France) from 11th to 15th June (2012), K(π,1) conjecture for Artin groups, Ann. Fac. Sci. Toulouse Math. (6) 23, no. 2 (2014).

[30] Reiner (V.).— Signed permutation statistics, Eur. J. Comb 14, p. 553-567 (1993). | MR | Zbl

[31] Salvetti (M.).— Topology of the complement of real hyperplanes in N , Invent. Math. 88, no. 3, p. 603-618 (1987). | MR | Zbl

[32] Salvetti (M.).— The homotopy type of Artin groups, Math. Res. Lett. 1, no. 5, p. 565-577 (1994). | MR | Zbl

[33] Solomon (L.).— The orders of the finite Chevalley groups, J. Algebra 3, p. 376-393 (1966). | MR | Zbl

[34] Spanier (E. H.).— Algebraic topology, McGraw-Hill Book Co., New York (1966). | MR | Zbl

[35] Steinberg (R.).— On Dickson’s theorem on invariants, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34, no. 3, p. 699-707 (1987). | MR | Zbl

[36] Tits (J.).— Le problème des mots dans les groupes de Coxeter, Symposia Mathematica (INDAM, Rome, 1967/68), vol. 1, Academic Press London, p. 175-185 (1969). | MR | Zbl

Cité par Sources :