Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Cohomology of affine Artin groups and applications

Authors: Filippo Callegaro, Davide Moroni and Mario Salvetti
Journal: Trans. Amer. Math. Soc. 360 (2008), 4169-4188
MSC (2000): Primary 20J06, 20F36
Published electronically: March 11, 2008
MathSciNet review: 2395168
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The result of this paper is the determination of the cohomology of Artin groups of type $ A_n, B_n$ and $ \tilde{A}_{n}$ with non-trivial local coefficients. The main result

is an explicit computation of the cohomology of the Artin group of type $ B_n$ with coefficients over the module $ \mathbb{Q}[q^{\pm 1},t^{\pm 1}].$ Here the first $ n-1$ standard generators of the group act by $ (-q)$-multiplication, while the last one acts by $ (-t)$-multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiro's lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type $ \tilde{A}_{n}$ as well as the cohomology of the classical braid group $ \mathrm{Br}_{n}$ with coefficients in the $ n$-dimensional representation presented in Tong, Yang, and Ma (1996). The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be $ K(\pi,1)$ spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.

References [Enhancements On Off] (What's this?)

  • [All02] D. Allcock, Braid pictures for Artin groups, Trans. Amer. Math. Soc. 354 (2002), no. 9, 3455-3474 (electronic). MR 1911508 (2003f:20053)
  • [Arn68] V. I. Arnol$ '$d, Braids of algebraic functions and cohomologies of swallowtails, Uspehi Mat. Nauk 23 (1968), no. 4 (142), 247-248. MR 0231828 (38:156)
  • [Bou68] N. Bourbaki, Éléments 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 0240238 (39:1590)
  • [Bri71] E. Brieskorn, Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe, Invent. Math. 12 (1971), 57-61. MR 0293615 (45:2692)
  • [Bro82] K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1982. MR 672956 (83k:20002)
  • [BS72] E. Brieskorn and K. Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245-271. MR 0323910 (48:2263)
  • [Cal05] F. Callegaro, On the cohomology of Artin groups in local systems and the associated Milnor fiber, J. Pure Appl. Algebra 197 (2005), no. 1-3, 323-332. MR 2123992 (2005k:20090)
  • [Cal06] F. Callegaro, The homology of the Milnor fiber for classical braid groups, Algebr. Geom. Topol. 6 (2006), 1903-1923 (electronic). MR 2263054
  • [CD95] R. Charney and M. W. Davis, The $ K(\pi,1)$-problem for hyperplane complements associated to infinite reflection groups, J. Amer. Math. Soc. 8 (1995), no. 3, 597-627. MR 1303028 (95i:52011)
  • [CMS] F. Callegaro, D. Moroni, and M. Salvetti, Cohomology of Artin braid groups of type $ \tilde{A}_n, {B}_n$ and applications, To be published in Geometry & Topology Monographs.
  • [CMS1] F. Callegaro, D. Moroni, and M. Salvetti, The $ K(\pi,1)$-problem for the affine Artin group of type $ \tilde{B}_n$ and its cohomology, (2006), to appear
  • [Coh76] F. R. Cohen, The homology of iterated loop spaces, Lecture Notes in Mathematics, vol. 533, The homology of $ C_{n+1}$-spaces, $ n\geq 0$, pp. 207-351, Springer-Verlag, Berlin, 1976. MR 0436146 (55:9096)
  • [CP03] R. Charney and D. Peifer, The $ K(\pi,1)$-conjecture for the affine braid groups, Comment. Math. Helv. 78 (2003), no. 3, 584-600. MR 1998395 (2004f:20067)
  • [Cri99] J. Crisp, Injective maps between Artin groups, Geometric group theory down under (Canberra, 1996), de Gruyter, Berlin, 1999, pp. 119-137. MR 1714842 (2001b:20064)
  • [CS98] D. C. Cohen and A. I. Suciu, Homology of iterated semidirect products of free groups, J. Pure Appl. Algebra 126 (1998), no. 1-3, 87-120. MR 1600518 (99e:20064)
  • [DCPS01] C. De Concini, C. Procesi, and M. Salvetti, Arithmetic properties of the cohomology of braid groups, Topology 40 (2001), no. 4, 739-751. MR 1851561 (2002f:20082)
  • [DCPSS99] C. De Concini, C. Procesi, M. Salvetti, and F. Stumbo, Arithmetic properties of the cohomology of Artin groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 4, 695-717. MR 1760537 (2001f:20078)
  • [DCS96] C. De Concini and M. Salvetti, Cohomology of Artin groups, Math. Res. Lett. 3 (1996), no. 2, 293-297. MR 1386847 (97b:52015)
  • [DCSS97] C. De Concini, M. Salvetti, and F. Stumbo, The top-cohomology of Artin groups with coefficients in rank-$ 1$ local systems over $ {\bf Z}$, Special issue on braid groups and related topics (Jerusalem, 1995), vol. 78, Topology Appl., no. 1-2, 1997, pp. 5-20. MR 1465022 (98h:20063)
  • [Del72] P. Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273-302. MR 0422673 (54:10659)
  • [Dũn83] Nguyen Viêt Dũng, The fundamental groups of the spaces of regular orbits of the affine Weyl groups, Topology 22 (1983), no. 4, 425-435. MR 715248 (85f:57001)
  • [Fre88] È. V. Frenkel$ '$, Cohomology of the commutator subgroup of the braid group, Funktsional. Anal. i Prilozhen. 22 (1988), no. 3, 91-92. MR 961774 (90h:20055)
  • [Fuk70] D. B. Fuks, Cohomology of the braid group $ {\rm mod} 2$, Funkcional. Anal. i Priložen. 4 (1970), no. 2, 62-73. MR 0274463 (43:226)
  • [Gor78] V. V. Gorjunov, The cohomology of braid groups of series $ C$ and $ D$ and certain stratifications, Funktsional. Anal. i Prilozhen. 12 (1978), no. 2, 76-77. MR 498905 (80g:32020)
  • [Hum90] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460 (92h:20002)
  • [KP02] R. P. Kent, IV and D. Peifer, A geometric and algebraic description of annular braid groups, Internat. J. Algebra Comput. 12 (2002), no. 1-2, 85-97, International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory (Lincoln, NE, 2000). MR 1902362 (2003f:20056)
  • [Lam94] S. S. F. Lambropoulou, Solid torus links and Hecke algebras of $ {B}$-type, Proceedings of the Conference on Quantum Topology (Manhattan, KS, 1993), World Sci. Publ., River Edge, NJ, 1994, pp. 225-245. MR 1309934 (96a:57020)
  • [Lan00] C. Landi, Cohomology rings of Artin groups, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11 (2000), no. 1, 41-65. MR 1797053 (2001j:20082)
  • [Mar96] N. S. Markaryan, Homology of braid groups with nontrivial coefficients, Mat. Zametki 59 (1996), no. 6, 846-854. MR 1445470 (98j:20047)
  • [Oko79] C. Okonek, Das $ K(\pi ,\,1)$-Problem für die affinen Wurzelsysteme vom Typ $ A\sb{n}$, $ C\sb{n}$, Math. Z. 168 (1979), no. 2, 143-148. MR 544701 (80i:32039)
  • [Rei93] V. Reiner, Signed permutation statistics, European J. Combin. 14 (1993), no. 6, 553-567. MR 1248063 (95e:05008)
  • [Sal87] M. Salvetti, Topology of the complement of real hyperplanes in $ {\bf C}\sp N$, Invent. Math. 88 (1987), no. 3, 603-618. MR 884802 (88k:32038)
  • [Sal94] M. Salvetti, The homotopy type of Artin groups, Math. Res. Lett. 1 (1994), no. 5, 565-577. MR 1295551 (95j:52026)
  • [Squ94] C. C. Squier, The homological algebra of Artin groups, Math. Scand. 75 (1994), no. 1, 5-43. MR 1308935 (95k:20059)
  • [SS97] M. Salvetti and F. Stumbo, Artin groups associated to infinite Coxeter groups, Discrete Math. 163 (1997), no. 1-3, 129-138. MR 1428564 (98d:20043)
  • [Sys01] I. Sysoeva, Dimension $ n$ representations of the braid group on $ n$ strings, J. Algebra 243 (2001), no. 2, 518-538. MR 1850645 (2002h:20054)
  • [Tit66] J. Tits, Normalisateurs de tores. I. Groupes de Coxeter étendus, J. Algebra 4 (1966), 96-116. MR 0206117 (34:5942)
  • [TYM96] Dian-Min Tong, Shan-De Yang, and Zhong-Qi Ma, A new class of representations of braid groups, Comm. Theoret. Phys. 26 (1996), no. 4, 483-486. MR 1456851 (98c:20073)
  • [Vaĭ78] F. V. Vaĭnšteĭn, The cohomology of braid groups, Funktsional. Anal. i Prilozhen. 12 (1978), no. 2, 72-73. MR 498903 (80g:32019)
  • [vdL83] H. van der Lek, The homotopy type of complex hyperplane complements, Ph.D. thesis, Katholieke Universiteit te Nijimegen, 1983.
  • [Vin71] È. B. Vinberg, Discrete linear groups that are generated by reflections, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1072-1112. MR 0302779 (46:1922)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20J06, 20F36

Retrieve articles in all journals with MSC (2000): 20J06, 20F36

Additional Information

Filippo Callegaro
Affiliation: Scuola Normale Superiore, dei Cavalieri, 7, Pisa, Italy

Davide Moroni
Affiliation: Dipartimento di Matematica “G.Castelnuovo”, A. Moro, 2, Roma, Italy – and – ISTI-CNR, Via G. Moruzzi, 3, Pisa, Italy

Mario Salvetti
Affiliation: Dipartimento di Matematica “L.Tonelli”, Largo B. Pontecorvo, 5, Pisa, Italy

Keywords: Affine Artin groups, twisted cohomology, group representations
Received by editor(s): June 20, 2006
Published electronically: March 11, 2008
Additional Notes: The third author is partially supported by M.U.R.S.T. 40%
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society