Kohno, Toshitake Série de Poincaré-Koszul associée aux groupes de tresses pures. (French) Zbl 0574.55009 Invent. Math. 82, 57-75 (1985). 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 Cited in 9 ReviewsCited in 49 Documents MSC: 55P62 Rational homotopy theory 17D10 Mal’tsev rings and algebras 32Q99 Complex manifolds 20F36 Braid groups; Artin groups Keywords:Mal’tsev completion; Poincaré series; pure braid groups; complement of a complex hypersurface; lower central series; holonomy algebra; free Lie algebra; mixed Hodge structure; nilpotent completion Citations:Zbl 0574.55010 PDF BibTeX XML Cite \textit{T. Kohno}, Invent. Math. 82, 57--75 (1985; Zbl 0574.55009) Full Text: DOI EuDML OpenURL References: [1] Aomoto, K.: On the acyclicity of free cober constructions I, II. Proc. Jap. Acad.53, 35-36 (1977) · Zbl 0385.57014 [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. 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