Série de Poincaré-Koszul associée aux groupes de tresses pures. (French) Zbl 0574.55009

Let X be the complement of a complex hypersurface in \({\mathbb{C}}P^ N\). The lower central series \(\Pi_ 1(X,*)=\Gamma_ 0\supset \Gamma_ 1\supset...\supset \Gamma_ n\supset..\). defines the Mal’tsev algebra of X, \(L_ X=\lim_{\leftarrow}(\Gamma_ 0/\Gamma_ j\otimes {\mathbb{Q}})\). A new Lie algebra \(J_ X\), called the holonomy algebra, is associated to X as follows. The dual of the cup product induces \(\delta\) : \(H_ 2(X,{\mathbb{Q}})\to H_ 1(X,{\mathbb{Q}})\otimes H_ 1(X,{\mathbb{Q}})\subset T(H_ 1(X,{\mathbb{Q}})\) which factorises into the free Lie algebra \(L(H_ 1(X,{\mathbb{Q}}))\). Then, \(J_ X\) is defined as the quotient of \(L(H_ 1(X,{\mathbb{Q}}))\) by the ideal generated by \(\delta (H_ 2(X,{\mathbb{Q}})).\)
Using the Sullivan theory and the mixed Hodge structure, the author proves that \(J^*_ H\cong L_ X\), when \(J^*_ X\) is the nilpotent completion of \(J_ X\). If \(X={\mathbb{C}}^{n+1}-\Lambda_ n\), where \(\Lambda_ n\) is the union of the hyperplanes \(H_{ij}\) \((x_ i=x_ j)\), then the author constructs a free resolution of \({\mathbb{Q}}\) by \(J_ X\)-modules and proves that there exists canonical isomorphisms \(H^ j(X,{\mathbb{Q}})\cong H^ j(\Pi_ 1(X),{\mathbb{Q}})\cong H^ j(J_ X,{\mathbb{Q}})\).
Reviewer: J.C.Thomas


55P62 Rational homotopy theory
17D10 Mal’tsev rings and algebras
32Q99 Complex manifolds
20F36 Braid groups; Artin groups


Zbl 0574.55010
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[1] Aomoto, K.: On the acyclicity of free cober constructions I, II. Proc. Jap. Acad.53, 35-36 (1977) · Zbl 0385.57014 · doi:10.3792/pjaa.53.35
[2] Aomoto, K.: Poincaré series of the holonomy Lie algebra attached to a configuration of lines. Nagoya University 1984 (Preprint)
[3] Arnold, V.I.: The cohomology ring of the colored braid group. Mat. Zametki5, 227-231 (1969); Math. Notes5, 138-140 (1969)
[4] Birman, J.: Braids, links, and mapping class groups. Ann. Math. Stud.82, Princeton, N.J.: Princeton University Press 1975 · Zbl 0305.57013
[5] Brieskorn, E.: Sur les groupes de tresses (d’après V.I. Arnold). Séminaire Bourbaki 24 e année 1971/1972. Lect. Notes, no. 317. Berlin: Springer 1973
[6] Chen, K.T.: Iterated integrals of differential forms and loop space cohomology. Ann. Math.97, 217-246 (1973) · Zbl 0227.58003 · doi:10.2307/1970846
[7] Deligne, P.: Théorie de Hodge II. Publ. Math. IHES40, 5-58 (1971) · Zbl 0219.14007
[8] Deligne, P.: Les immeubles des groupes de tresses généralisés. Invent. Math.17, 273-302 (1972) · Zbl 0238.20034 · doi:10.1007/BF01406236
[9] Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math.29, 245-274 (1975) · Zbl 0312.55011 · doi:10.1007/BF01389853
[10] Friedlander, E., Griffiths, P., Morgan, J.: Homotopy theory of differential forms. Seminario di geometria, Firenze 1972
[11] Greub, W., Halperin, S., Vanstone, R.: Connections, curvature and cohomology III. New York: Academic Press 1976 · Zbl 0372.57001
[12] Hain, R.M.: The de Rham homotopy theory of complex algebraic varieties. University of Utah 1984 (Preprint) · Zbl 0637.55006
[13] Hopf, H.: Fundamentalgruppe und zweite Bettische Gruppe. Comment. Math. Helv.14, 257-309 (1941-42) · Zbl 0027.09503 · doi:10.1007/BF02565622
[14] Hilton, J.P., Stammbach, U.: A course in homological algebra. GTM. Berlin-Heidelberg-New York: Springer 1970 · Zbl 0863.18001
[15] Kohno, T.: Differential forms and the fundamental group of the complement of hypersurfaces. Proc. Pure Math. Am. Math. Soc.40, 655-662 (1983) · Zbl 0519.57001
[16] Kohno, T.: On the holonomy Lie algebra and the nilpotent completion of the fundamental group of the complement of hypersurfaces. Nagoya Math. J.92, 21-37 (1983) · Zbl 0503.57001
[17] Labute, J.P.: On the descending central series of groups with a single defining relation. J. Algebra14, 16-23 (1970) · Zbl 0198.34601 · doi:10.1016/0021-8693(70)90130-4
[18] Lyndon, R.: Cohomology theory of groups with a single defining relation. Ann. Math.52, 650-665 (1950) · Zbl 0039.02302 · doi:10.2307/1969440
[19] Malcev, A.: Nilpotent groups without torsion. Izv. Akad. Nauk SSSR13, 201-212 (1949)
[20] Morgan, J.: The algebraic topology of smooth algebraic varieties. Publ. Math. IHES48, 137-204 (1978) · Zbl 0401.14003
[21] Magnus, W., Karrass, A., Solitar, D.: Combinatorial group theory. New York: John Wiley 1966 · Zbl 0138.25604
[22] Orlik, P., Solomon, L.: Combinatorics and the topology of complement of hyperplanes. Invent. Math.56, 167-189 (1980) · Zbl 0432.14016 · doi:10.1007/BF01392549
[23] Sullivan, D.: Infinitesimal computations in topology. Publ. Math. IHES47, 269-331 (1977) · Zbl 0374.57002
[24] Terao, H.: On Betti numbers of complement of hyperplanes. Publ. Res. Inst. Math. Sci. Kyoto17, 657-663 (1981) · Zbl 0484.14004 · doi:10.2977/prims/1195185267
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