×

zbMATH — the first resource for mathematics

Homology computations for complex braid groups. (English) Zbl 1302.20049
Complex braid groups are defined as follows: let \(W\) be a complex reflection group, i.e. a finite group generated by finite order endomorphisms of \(\mathrm{GL}_r(\mathbb C)\) that leave invariant some hyperplane in \(\mathbb C^r\). Let \(\mathcal A\) be the (central) hyperplane arrangement associated to the reflections of \(W\), and let \(X=\mathbb C^r\setminus\bigcup\mathcal A\) be the corresponding hyperplane complement. The generalized braid group \(B=\pi_1(X/W)\) is an extension of \(W\) by \(P=\pi_1(X)\). These braid groups are parametrized by three integers \(B(de,e,r)\) with 34 exceptional cases \(B_4,\ldots,B_{37}\). The full classification of these groups is not yet fully understood.
The authors give some partial classifications of these braid groups, some specific calculations of their homology in low degrees and some results about the stability behavior of their homology. Here are some examples of their results:
Theorem 6.4. Let \(B=B(e,e,r)\) and \(r\geq 3\). Then \[ H_2(B;\mathbb Z)\cong\begin{cases}\mathbb Z/e &\text{for }r=3,\\ \mathbb Z/e\times\mathbb Z/2 &\text{for }r=4\text{ and odd }e,\\ \mathbb Z/e\times(\mathbb Z/2)^2 &\text{for }r=4\text{ and even }e,\text{ and}\\ \mathbb Z/e\times\mathbb Z/2 &\text{for }r\geq 5.\end{cases} \] As for stability results, we have the following Theorem. Let \(p\) be an odd prime, the homology group \(H_*(B(2e,e,\infty);\mathbb F_p)\) is isomorphic to: \[ \lim_{r\to\infty}H_*(B(2e,e,r);\mathbb F_p)\cong\mathbb F_p[w_1,\overline y_1,\overline y_2,\ldots]\otimes\Lambda[\overline x_0,\overline x_1,\ldots], \] where \(\dim(w_1)=1\), \(\dim(\overline x_i)=2p^i-1\) and \(\dim(\overline y_i)=2p^i-2\). Moreover, the canonical morphism \[ H_i(B(2e,e,r);\mathbb F_p)\to H_i(B(2e,e,\infty);\mathbb F_p) \] is an isomorphism for \(r>(i-1)\frac{p}{p-1}+2\).

MSC:
20J05 Homological methods in group theory
20J06 Cohomology of groups
20F36 Braid groups; Artin groups
20F55 Reflection and Coxeter groups (group-theoretic aspects)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bannai, E.: Fundamental groups of the spaces of regular orbits of the finite uni- tary reflection groups of dimension 2. J. Math. Soc. Japan 28, 447-454 (1976) · Zbl 0326.57015
[2] Bessis, D., Corran, R.: Non-crossing partitions of type (e, e, r). Adv. Math. 202, 1-49 (2006) · Zbl 1128.20024
[3] Bessis, D.: Finite complex reflection arrangements are K(\pi , 1). · Zbl 1372.20036
[4] Bessis, D., Michel, J.: Explicit presentations for exceptional braid groups. Experiment. Math. 13, 257-266 (2004) · Zbl 1092.20033
[5] Broué, M., Malle, G., Rouquier, R.: Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math. 500, 127-190 (1998) · Zbl 0921.20046
[6] Bonnafé, C.: Une (nouvelle?) construction du groupe de réflexion complexe G31. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 45 (93), 133-143 (2002) · Zbl 1084.20509
[7] Brieskorn, E.: Sur les groupes de tresses [d’apr‘es V. I. Arnol’d]. In: Séminaire Bour- baki, 24‘eme année (1971/1972), exp. 401, Lecture Notes in Math. 317, Springer, Berlin, 21-44 (1973) · Zbl 0277.55003
[8] Brown, K. S.: Cohomology of Groups. Grad. Texts in Math. 87, Springer, New York (1982) · Zbl 0584.20036
[9] Callegaro, F.: The homology of the Milnor fiber for classical braid groups. Algebr. Geom. Topol. 6, 1903-1923 (2006) · Zbl 1166.20044
[10] Callegaro, F., Moroni, D., Salvetti, M.: Cohomology of affine Artin groups and applica- tions. Trans. Amer. Math. Soc. 360, 4169-4188 (2008) · Zbl 1191.20056
[11] Charney, R., Meier, J., Whittlesey, K.: Bestvina’s normal form complex and the ho- mology of Garside groups. Geom. Dedicata 105, 171-188 (2004) · Zbl 1064.20044
[12] Cohen, F. R.: The homology of Cn+1-spaces, n \geq 0. In: Homology of Iterated Loop Spaces, Lecture Notes in Math. 533, Springer, 207-353 (1976) · Zbl 0334.55009
[13] Cohen, F. R.: Artin’s braid groups, classical homotopy theory, and sundry other cu- riosities. In: Braids (Santa Cruz, CA, 1986), Contemp. Math. 78, Amer. Math. Soc., Providence, RI, 167-206 (1988) · Zbl 0577.20026
[14] Corran, R., Picantin, M.: A new Garside structure for the braid groups of type (e, e, r). J. London Math. Soc. 84, 689-711 (2011) · Zbl 1239.20042
[15] Crisp, J.: Injective maps between Artin groups. In: Geometric Group Theory Down Under (Canberra, 1996), de Gruyter, Berlin, 119-137 (1999) · Zbl 1001.20034
[16] Deligne, P.: Les immeubles des groupes de tresses généralisés. Invent. Math. 17, 273- 302 (1972) · Zbl 0238.20034
[17] Dehornoy, P., Lafont, Y.: Homology of Gaussian groups. Ann. Inst. Fourier (Grenoble) 53, 489-540 (2003) · Zbl 1100.20036
[18] Digne, F., Marin, I., Michel, J.: The center of pure complex braid groups. J. Algebra 347, 206-213 (2011) · Zbl 1241.20039
[19] Dehornoy, P., Paris, L.: Gaussian groups and Garside groups, two generalisations of Artin groups. Proc. London Math. Soc. (3) 79, 569-604 (1999) · Zbl 1030.20021
[20] De Concini, C., Procesi, C., Salvetti, M.: Arithmetic properties of the cohomology of braid groups. Topology 40, 739-751 (2001) · Zbl 0999.20046
[21] De Concini, C., Procesi, C., Salvetti, M., Stumbo, F.: Arithmetic properties of the coho- mology of Artin groups. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28, 695-717 (1999) · Zbl 0973.20025
[22] Fadell, E., Neuwirth, L.: Configuration spaces. Math. Scand. 10, 111-118 (1962) · Zbl 0136.44104
[23] Fuks, D. B.: Cohomology of the braid group mod 2. Funct. Anal. Appl. 4, 143-151 (1970) · Zbl 0222.57031
[24] Fuks, D. B.: Quillenization and bordism. Funktsional. Anal. i Prilozhen. 8, no. 1, 36-42 (1974) (in Russian) · Zbl 0324.57024
[25] Gorjunov, V. V.: Cohomology of braid groups of series C and D. Trudy Moskov. Mat. Obshch. 42, 234-242 (1981) (in Russian) · Zbl 0547.55016
[26] Kent IV, R. P., Peifer, D.: A geometric and algebraic description of annular braid groups. Internat. J. Algebra Comput. 12, 85-97 (2002) · Zbl 1010.20024
[27] Kobayashi, Y.: Complete rewriting systems and homology of monoid algebras. J. Pure Appl. Algebra 65, 263-275 (1990) · Zbl 0711.20035
[28] Lehrer, G. I.: Poincaré polynomials for unitary reflection groups. Invent. Math. 120, 411-425 (1995) · Zbl 0831.20049
[29] Lehrer, G. I.: Rational points and cohomology of discriminant varieties. Adv. Math. 186, 229-250 (2004) · Zbl 1077.14025
[30] Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. Interscience Publ., New York (1966) · Zbl 0138.25604
[31] Michel, J.: Homepage of the development version of the GAP part of CHEVIE.
[32] Nakamura, T.: A note on the K(\pi , 1) property of the orbit space of the unitary re- flection group G(m, l, n). Sci. Papers College Arts Sci. Univ. Tokyo 33, 1-6 (1983) · Zbl 0524.20027
[33] Orlik, P., Terao, H.: Arrangements of Hyperplanes. Grundlehren Math. Wiss. 300, Springer, Berlin (1992) · Zbl 0757.55001
[34] Paris, L.: Artin groups of spherical type up to isomorphism. J. Algebra 281, 666-678 (2004) · Zbl 1080.20033
[35] Picantin, M.: Petits groupes gaussiens. Ph.D. thesis, Univ. de Caen (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.