cnrs   Université de Toulouse
With cedram.org

Table of contents for this issue | Previous article | Next article
Luis Paris
$K(\pi ,1)$ conjecture for Artin groups
Annales de la faculté des sciences de Toulouse Sér. 6, 23 no. 2: Numéro Spécial à l’occasion de la conférence Arrangements in Pyrénées, Pau 11-15 juin 2012 (2014), p. 361-415, doi: 10.5802/afst.1411
Article PDF | Reviews MR 3205598 | Zbl 06297897

Résumé - Abstract

The purpose of this paper is to put together a large amount of results on the $K(\pi ,1)$ conjecture for Artin groups, and to make them accessible to non-experts. Firstly, this is a survey, containing basic definitions, the main results, examples and an historical overview of the subject. But, it is also a reference text on the topic that contains proofs of a large part of the results on this question. Some proofs as well as few results are new. Furthermore, the text, being addressed to non-experts, is as self-contained as possible.

Bibliography

[1] Abramenko (P.), Brown (K. S.).— Buildings. Theory and applications. Graduate Texts in Mathematics, 248. Springer, New York (2008).  MR 2439729 |  Zbl 1214.20033
[2] Arnol’d (V. I.).— Certain topological invariants of algebraic functions. Trudy Moskov. Mat. Obšč. 21, p. 27-46 (1970).  MR 274462 |  Zbl 0208.24003
[3] Bourbaki (N.).— Eléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV : Groupes de Coxeter et systèmes de Tits. Chapitre V : Groupes engendrés par des réflexions. Chapitre VI : Systèmes de racines. Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris (1968).  MR 240238 |  Zbl 0186.33001
[4] Brieskorn (E.).— Sur les groupes de tresses [d’après V. I. Arnol’d]. Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, p. 21-44. Lecture Notes in Math., Vol. 317, Springer, Berlin (1973). Numdam |  MR 422674 |  Zbl 0277.55003
[5] Brieskorn (E.), Saito (K.).— Artin-Gruppen und Coxeter-Gruppen. Invent. Math. 17, p. 245-271 (1972).  MR 323910 |  Zbl 0243.20037
[6] Brown (K. S.).— Cohomology of groups. Graduate Texts in Mathematics, 87. Springer-Verlag, New York-Berlin (1982).  MR 672956 |  Zbl 0584.20036
[7] 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
[8] Callegaro (F.).— The homology of the Milnor fiber for classical braid groups. Algebr. Geom. Topol. 6, p. 1903-1923 (2006).  MR 2263054 |  Zbl 1166.20044
[9] 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
[10] Callegaro (F.), Moroni (D.), Salvetti (M.).— Cohomology of Artin groups of type $\tilde{A}_n, B_n$ and applications. Groups, homotopy and configuration spaces, 85-104, Geom. Topol. Monogr., 13, Geom. Topol. Publ., Coventry (2008).  MR 2508202 |  Zbl 1143.20031
[11] Callegaro (F.), Moroni (D.), Salvetti (M.).— The $K(\pi ,1)$ problem for the affine Artin group of type $\tilde{B}_n$ and its cohomology. J. Eur. Math. Soc. (JEMS) 12, no. 1, p. 1-22 (2010).  MR 2578601 |  Zbl 1190.20042
[12] Callegaro (F.), Salvetti (M.).— Integral cohomology of the Milnor fibre of the discriminant bundle associated with a finite Coxeter group. C. R. Math. Acad. Sci. Paris 339, no. 8, p. 573-578 (2004).  MR 2111354 |  Zbl 1059.32008
[13] Charney (R.), Davis (M. W.).— The $K(\pi ,1)$-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
[14] Charney (R.), Davis (M. W.).— Finite $K(\pi ,1)$’s for Artin groups. Prospects in topology (Princeton, NJ, 1994), p. 110-124, Ann. of Math. Stud., 138, Princeton Univ. Press, Princeton, NJ (1995).  MR 1368655 |  Zbl 0930.55006
[15] Charney (R.), Meier (J.), Whittlesey (K.).— Bestvina’s normal form complex and the homology of Garside groups. Geom. Dedicata 105, p. 171-188 (2004).  MR 2057250 |  Zbl 1064.20044
[16] Charney (R.), Peifer (D.).— The $K(\pi ,1)$-conjecture for the affine braid groups. Comment. Math. Helv. 78, no. 3, 584-600 (2003).  MR 1998395 |  Zbl 1066.20043
[17] Cohen (F. R.).— The homology of $C_{n+1}$-spaces, $n \ge 0$. Lecture Notes in Math. 533, p. 207-353, Springer-Verlag, Berlin-New York (1976).
[18] Coxeter (H. S. M.).— Discrete groups generated by reflections. Ann. of Math. (2) 35, no. 3, p. 588-621 (1934).  MR 1503182 |  Zbl 0010.01101
[19] Coxeter (H. S. M.).— The complete enumeration of finite groups of the form $R_i^2 = (R_i R_j)^{k_{i, j}} = 1$. J. London Math. Soc. 10, p. 21-25 (1935).  Zbl 0010.34202
[20] De Concini (C.), Procesi (C.), Salvetti (M.).— Arithmetic properties of the cohomology of braid groups. Topology 40, no. 4, p. 739-751 (2001).  MR 1851561 |  Zbl 0999.20046
[21] De Concini (C.), Procesi (C.), Salvetti (M.), Stumbo (F.).— Arithmetic properties of the cohomology of Artin groups. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28, no. 4, p. 695-717 (1999). Numdam |  MR 1760537 |  Zbl 0973.20025
[22] De Concini (C.), Salvetti (M.).— Stability for the cohomology of Artin groups. Adv. Math. 145, no. 2, p. 291-305 (1999).  MR 1704578 |  Zbl 0982.20036
[23] De Concini (C.), Salvetti (M.).— Cohomology of Coxeter groups and Artin groups. Math. Res. Lett. 7, no. 2-3, p. 213-232 (2000).  MR 1764318 |  Zbl 0972.20030
[24] De Concini (C.), Salvetti (M.), Stumbo (F.).— The top-cohomology of Artin groups with coefficients in rank-$1$ local systems over ${\sc Z}$. Special issue on braid groups and related topics (Jerusalem, 1995). Topology Appl. 78, no. 1-2, p. 5-20 (1997).  MR 1465022 |  Zbl 0878.55003
[25] Dehornoy (P.), Lafont (Y.).— Homology of Gaussian groups. Ann. Inst. Fourier (Grenoble) 53, no. 2, p. 489-540 (2003). Cedram |  MR 1990005 |  Zbl 1100.20036
[26] Deligne (P.).— Les immeubles des groupes de tresses généralisés. Invent. Math. 17, p. 273-302 (1972).  MR 422673 |  Zbl 0238.20034
[27] Dobrinskaya (N. È.).— The Arnol’d-Thom-Pham conjecture and the classifying space of a positive Artin monoid. (Russian) Uspekhi Mat. Nauk 57 (2002), no. 6(348), 181-182. Translation in Russian Math. Surveys 57, no. 6, p. 1215-1217 (2002).  MR 1991872 |  Zbl 1050.55008
[28] Ellis (G.), Sköldberg (E.).— The $K(\pi ,1)$ conjecture for a class of Artin groups. Comment. Math. Helv. 85, no. 2, p. 409-415 (2010).  MR 2595184 |  Zbl 1192.55011
[29] Fox (R.), Neuwirth (L.).— The braid groups. Math. Scand. 10, 119-126 (1962).  MR 150755 |  Zbl 0117.41101
[30] Fuks (D. B.).— Cohomology of the braid group mod 2. Funkcional. Anal. i Priložen. 4 (1970), no. 2, 62-73. Translation in Functional Anal. Appl. 4, p. 143-151 (1970).  MR 274463 |  Zbl 0222.57031
[31] Godelle (E.), Paris (L.).— $K(\pi ,1)$ and word problems for infinite type Artin-Tits groups, and applications to virtual braid groups. Math. Z. 272, no. 3, p. 1339-1364 (2012).  MR 2995171 |  Zbl pre06116871
[32] Hatcher (A.).— Algebraic topology. Cambridge University Press, Cambridge (2002).  MR 1867354 |  Zbl 1044.55001
[33] Hendriks (H.).— Hyperplane complements of large type. Invent. Math. 79, no. 2, p. 375-381 (1985).  MR 778133 |  Zbl 0564.57016
[34] Landi (C.).— Cohomology rings of Artin groups. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11, no. 1, p. 41-65 (2000).  MR 1797053 |  Zbl 0966.55012
[35] van der Lek (H.).— The homotopy type of complex hyperplane complements. Ph. D. Thesis, Nijmegen (1983).
[36] McCammond (J.), Sulway (R.).— Artin groups of Euclidean type. Preprint, arXiv:1312.7770
[37] Michel (J.).— A note on words in braid monoids. J. Algebra 215, no. 1, p. 366-377 (1999).  MR 1684142 |  Zbl 0937.20017
[38] Okonek (C.).— Das $K(\pi ,\,1)$-Problem für die affinen Wurzelsysteme vom Typ $A_{n}$, $C_{n}$. Math. Z. 168, no. 2, p. 143-148 (1979).  MR 544701 |  Zbl 0427.14001
[39] Orlik (P.), Terao (H.).— Arrangements of hyperplanes. Grundlehren der Mathematischen Wissenschaften, 300. Springer-Verlag, Berlin (1992).  MR 1217488 |  Zbl 0757.55001
[40] Ozornova (V.).— Factorability, Discrete Morse Theory and a Reformulation of $K(\pi ,1)$-conjecture. Ph. D. Thesis, Bonn (2013).
[41] Paris (L.).— Universal cover of Salvetti’s complex and topology of simplicial arrangements of hyperplanes. Trans. Amer. Math. Soc. 340, no. 1, p. 149-178 (1993).  MR 1148044 |  Zbl 0805.57018
[42] Paris (L.).— Artin monoids inject in their groups. Comment. Math. Helv. 77, no. 3, p. 609-637 (2002).  MR 1933791 |  Zbl 1020.20026
[43] Salvetti (M.).— Topology of the complement of real hyperplanes in $\mathbb{C}^N$. Invent. Math. 88, no. 3, p. 603-618 (1987).  MR 884802 |  Zbl 0594.57009
[44] Salvetti (M.).— On the homotopy theory of complexes associated to metrical-hemisphere complexes. Discrete Math. 113, no. 1-3, p. 155-177 (1993).  MR 1212876 |  Zbl 0774.52007
[45] Salvetti (M.).— The homotopy type of Artin groups. Math. Res. Lett. 1, no. 5, p. 565-577 (1994).  MR 1295551 |  Zbl 0847.55011
[46] Salvetti (M.), Stumbo (F.).— Artin groups associated to infinite Coxeter groups. Discrete Math. 163, no. 1-3, p. 129-138 (1997).  MR 1428564 |  Zbl 0871.05031
[47] Segal (G.).— Configuration-spaces and iterated loop-spaces. Invent. Math. 21, p. 213-221 (1973).  MR 331377 |  Zbl 0267.55020
[48] Settepanella (S.).— A stability-like theorem for cohomology of pure braid groups of the series A, B and D. Topology Appl. 139, no. 1-3, p. 37-47 (2004).  MR 2051096 |  Zbl 1064.20052
[49] Settepanella (S.).— Cohomology of pure braid groups of exceptional cases. Topology Appl. 156, no. 5, p. 1008-1012 (2009).  MR 2498934 |  Zbl 1195.20055
[50] Spanier (E. H.).— Algebraic topology. Corrected reprint. Springer-Verlag, New York-Berlin (1981).  MR 666554 |  Zbl 0810.55001
[51] Tits (J.).— Le problème des mots dans les groupes de Coxeter. Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1, p. 175-185, Academic Press, London (1969).  MR 254129 |  Zbl 0206.03002
[52] Tits (J.).— Groupes et géométries de Coxeter. In Wolf Prize in Mathematics, Vol. 2, S. S. Chern and F. Hirzebruch, eds., World Scientific Publishing, River Edge, NJ, p. 740-754 (2001).
[53] Vaǐnšteǐn (F. V.).— The cohomology of braid groups. Funktsional. Anal. i Prilozhen. 12 (1978), no. 2, p. 72-73. Translation in Functional Anal. Appl. 12, no. 2, p. 135-137 (1978).  MR 498903
[54] Vinberg (E. B.).— Discrete linear groups that are generated by reflections. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 35, p. 1072-1112 (1971).  MR 302779 |  Zbl 0247.20054
Search for an article
Search within the site
top