# zbMATH — the first resource for mathematics

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

##### MSC:
 55P62 Rational homotopy theory 17D10 Mal’tsev rings and algebras 32Q99 Complex manifolds 20F36 Braid groups; Artin groups
Full Text:
##### References:
  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  Aomoto, K.: Poincaré series of the holonomy Lie algebra attached to a configuration of lines. Nagoya University 1984 (Preprint)  Arnold, V.I.: The cohomology ring of the colored braid group. Mat. Zametki5, 227-231 (1969); Math. Notes5, 138-140 (1969)  Birman, J.: Braids, links, and mapping class groups. Ann. Math. Stud.82, Princeton, N.J.: Princeton University Press 1975 · Zbl 0305.57013  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  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  Deligne, P.: Théorie de Hodge II. Publ. Math. IHES40, 5-58 (1971) · Zbl 0219.14007  Deligne, P.: Les immeubles des groupes de tresses généralisés. Invent. Math.17, 273-302 (1972) · Zbl 0238.20034 · doi:10.1007/BF01406236  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  Friedlander, E., Griffiths, P., Morgan, J.: Homotopy theory of differential forms. Seminario di geometria, Firenze 1972  Greub, W., Halperin, S., Vanstone, R.: Connections, curvature and cohomology III. New York: Academic Press 1976 · Zbl 0372.57001  Hain, R.M.: The de Rham homotopy theory of complex algebraic varieties. University of Utah 1984 (Preprint) · Zbl 0637.55006  Hopf, H.: Fundamentalgruppe und zweite Bettische Gruppe. Comment. Math. Helv.14, 257-309 (1941-42) · Zbl 0027.09503 · doi:10.1007/BF02565622  Hilton, J.P., Stammbach, U.: A course in homological algebra. GTM. Berlin-Heidelberg-New York: Springer 1970 · Zbl 0863.18001  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  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  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  Lyndon, R.: Cohomology theory of groups with a single defining relation. Ann. Math.52, 650-665 (1950) · Zbl 0039.02302 · doi:10.2307/1969440  Malcev, A.: Nilpotent groups without torsion. Izv. Akad. Nauk SSSR13, 201-212 (1949)  Morgan, J.: The algebraic topology of smooth algebraic varieties. Publ. Math. IHES48, 137-204 (1978) · Zbl 0401.14003  Magnus, W., Karrass, A., Solitar, D.: Combinatorial group theory. New York: John Wiley 1966 · Zbl 0138.25604  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  Sullivan, D.: Infinitesimal computations in topology. Publ. Math. IHES47, 269-331 (1977) · Zbl 0374.57002  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
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.