Lüdde, Mirko Generalised Magnus modules over the braid group. (English) Zbl 0859.20030 Math. Ann. 306, No. 3, 555-569 (1996). The present article considers the construction of homology modules over the braid group. A well-known procedure of W. Magnus, restated by J. Birman, is shown to be equivalent to considering the homology \(H_1(F,M)\) of the free group \(F\) with certain coefficient modules \(M\) as braid module. The main result of the work is the construction of a free resolution \(C\) of the integers over an iterated semidirect product of free groups. The complexes \(P\otimes C\) and their homology for certain coefficient modules \(P\) then carry the structure of a braid module. These can be regarded as generalisations of the Burau- and Magnus-modules. The resolution is further used to show that the holonomy representations of the braid group and of the Hecke algebra constructed by topological means by R. J. Lawrence belong to this class of modules. Reviewer: M.Lüdde (Berlin) Cited in 1 Review MSC: 20F36 Braid groups; Artin groups 32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) Keywords:homology modules; braid modules; free resolutions; semidirect products of free groups; holonomy representations; braid groups; Hecke algebras PDFBibTeX XMLCite \textit{M. Lüdde}, Math. Ann. 306, No. 3, 555--569 (1996; Zbl 0859.20030) Full Text: DOI arXiv EuDML References: [1] [Ati90] Michael F. Atiyah: Representations of braid groups. Geometry of Low-Dimensional Manifolds: 2, London Math. Soc. Lect. Notes Ser., vol. 151, Cambridge U.P., 1990, pp. 115-122 [2] [Bir74] Joan Birman: Braids, links and mapping class groups. Ann. Math. Stud., vol. 82, Princeton U.P., 1974 [3] [Bro82] Kenneth S. Brown: Cohomology of groups. GTM, vol. 87, Springer, 1982 [4] [CL92] Florin Constantinescu, Mirko Lüdde: Braid modules. J. Phys. A: Math. Gen.25 (1992), L1273-L1280 · Zbl 0790.20010 · doi:10.1088/0305-4470/25/23/002 [5] [Jon91] Vaughan F.R. Jones: Subfactors and knots. Reg. Conf. Ser. in Math., vol. 80, Amer. Math. Soc., 1991 [6] [Law90] Ruth J. Lawrence: Homological representations of the Hecke algebra. Comm. Math. Phys.135 (1990), 141-191 · Zbl 0716.20022 · doi:10.1007/BF02097660 [7] [Law93] Ruth J. Lawrence: A functorial approach to the one-variable Jones polynomial. J. Diff. Geom.37 (1993), 689-710 · Zbl 0795.57005 [8] [Lüd92] Mirko Lüdde: Treue Darstellungen der Zopfgruppe und einige Anwendungen. Dissertation, Physikalisches Institut, Universität Bonn, November 1992, Preprint IR-92-49 [9] [Lüd95] Mirko Lüdde: Notes on generalised Magnus modules over the braid group. Preprint SFB288: Differentialgeometrie und Quantenphysik170 (1995), available at http://www.math.tu-berlin.de [10] [Mag74] Wilhelm Magnus: Braid groups: A survey. Lecture Notes in Mathematics, vol. 372, Springer, 1974, pp. 463-487 · doi:10.1007/BFb0065203 [11] [Mil73] John W. Milnor: Morse theory. Ann. of Math. Stud., vol. 51, Princeton U.P., 1973 [12] [Whi78] George W. Whitehead: Elements of homotopy theory. GTM, vol. 61, Springer, N.Y., 1978 · Zbl 0406.55001 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.