## 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

Zbl 0574.55010
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### References:

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