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

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.

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.

@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},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 23},
     number = {2},
     year = {2014},
     pages = {267-296},
     doi = {10.5802/afst.1407},
     zbl = {06297893},
     language = {en},
     url = {http://www.numdam.org/item/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, Serie 6, Volume 23 (2014) no. 2, pp. 267-296. doi : 10.5802/afst.1407. http://www.numdam.org/item/AFST_2014_6_23_2_267_0/

[1] Artin (E.).— Theorie des zöpfe, Abh. Math. Sem. Univ. Hamburg 4, p. 47-72 (1925). | MR 3069440

[2] Brieskorn (E.).— Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe, Invent. Math. 12, p. 57-61 (1971). | MR 293615 | Zbl 0204.56502

[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 422674 | Zbl 0277.55003

[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 1324339 | Zbl 0584.20036

[5] Björner (A.), Ziegler (G. M.).— Combinatorial stratification of complex arrangements, J. Amer. Math. Soc. 5, no. 1, p. 105-149 (1992). | MR 1119198 | Zbl 0754.52003

[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 2123992 | Zbl 1109.20027

[7] Callegaro (F.).— The homology of the Milnor fiber for classical braid groups, Algebr. Geom. Topol. 6, p. 1903-1923 (electronic) (2006). | MR 2263054 | Zbl 1166.20044

[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 3094502 | Zbl pre06214611

[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 1303028 | Zbl 0833.51006

[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 2395168 | Zbl 1191.20056

[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 2578601 | Zbl 1190.20042

[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 1851561 | Zbl 0999.20046

[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 1386847 | Zbl 0870.57004

[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 1465022 | Zbl 0878.55003

[16] Deligne (P.).— Les immeubles des groupes de tresses généralisés, Invent. Math. 17, p. 273-302 (1972). | MR 422673 | Zbl 0238.20034

[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 1500882

[18] Fox (R.), L. Neuwirth (L.).— The braid groups, Math. Scand. 10, p. 119-126 (1962). | MR 150755 | Zbl 0117.41101

[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 961774 | Zbl 0675.20042

[20] Fuks (D. B.).— Cohomology of the braid group mod 2, Funct. Anal. Appl. 4, no. 2, p. 143-151 (1970). | MR 274463 | Zbl 0222.57031

[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 1020668 | Zbl 0717.33009

[22] Humphreys (J. E.).— Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge (1990). | MR 1066460 | Zbl 0725.20028

[23] Markaryan (N. S.).— Homology of braid groups with nontrivial coefficients, Mat. Zametki 59, no. 6, p. 846-854, 960 (1996). | MR 1445470 | Zbl 0884.55016

[24] Matsumoto (H.).— Générateurs et relations des groupes de Weyl généralisés, C. R. Acad. Sci. Paris 258, p. 3419-3422 (1964). | MR 183818 | Zbl 0128.25202

[25] Milnor (J.).— Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J. (1968). | MR 239612 | Zbl 0184.48405

[26] Milnor (J.).— Introduction to algebraic K-theory, Princeton University Press, Princeton, N.J., Annals of Mathematics Studies, No. 72 (1971). | MR 349811 | Zbl 0237.18005

[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 207802 | Zbl 0362.20023

[28] Orlik (P.), Terao (H.).— Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin (1992). | MR 1217488 | Zbl 0757.55001

[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 1248063 | Zbl 0793.05005

[31] Salvetti (M.).— Topology of the complement of real hyperplanes in N , Invent. Math. 88, no. 3, p. 603-618 (1987). | MR 884802 | Zbl 0594.57009

[32] Salvetti (M.).— The homotopy type of Artin groups, Math. Res. Lett. 1, no. 5, p. 565-577 (1994). | MR 1295551 | Zbl 0847.55011

[33] Solomon (L.).— The orders of the finite Chevalley groups, J. Algebra 3, p. 376-393 (1966). | MR 199275 | Zbl 0151.02003

[34] Spanier (E. H.).— Algebraic topology, McGraw-Hill Book Co., New York (1966). | MR 210112 | Zbl 0477.55001

[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 927606 | Zbl 0656.20052

[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 254129 | Zbl 0206.03002