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The cohomology and fundamental group of a hyperplane complement. (English) Zbl 0697.55013

Singularities, Proc. IMA Participating Inst. Conf., Iowa City/Iowa 1986, Contemp. Math. 90, 55-72 (1989).
[For the entire collection see Zbl 0668.00006.]
Let \(A=H_ 1,...,H_ n\) be an arrangement of hyperplanes through 0 in \(C^{\ell}\), with complement \(M=C^{\ell}-\cup^{n}_{i-1}H_ i\). The cohomology algebra \(H^*(M)\) is generated by degree one elements \(\omega_ 1,...,\omega_ n\) dual to the hyperplanes. The relations among the \(\omega_ i\) have been described explicitly in terms of the intersection lattice of the arrangement [P. Orlik and L. Solomon, Invent. Math. 56, 167-189 (1980; Zbl 0432.14016)], and this structure has been studied extensively. In this article we describe a relatively new technique for the analysis of these algebras, the Sullivan one-minimal model. The theory of minimal models is well suited algebraically to the study of arrangements, because M is a formal space, in the sense of D. Sullivan, [Inst. Haut. Étud. Sci., Publ. Math. 47, 269-331 (1977; Zbl 0374.57002)], and also because the presentation of \(H^*(M)\) is so nice. The topological consequences are somewhat limited, however, because \(\pi_ 1(M)\) is so complicated. One does obtain information on the factors of the lower central series of the fundamental group \(\pi_ 1(M)\)- these results have been described at length in other papers [the author, “The minimal model of the complement of an arrangement of hyperplanes”, Trans. Am. Math. Soc. (to appear); the author and R. Randell, Contemp. Math. 78, 217-228 (1988; Zbl 0669.20028); T. Kohno, Invent. Math. 82, 57-75 (1985; Zbl 0574.55009)]. In this paper we view the minimal model mainly as a tool for distinguishing the cohomology algebras associated with combinatorially distinct arrangements.

MSC:

55P62 Rational homotopy theory
57R19 Algebraic topology on manifolds and differential topology