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Topological and differentiable structures of the complement of an arrangement of hyperplanes. (English) Zbl 0795.57012
Greene, Robert (ed.) et al., Differential geometry. Part 2: Geometry in mathematical physics and related topics. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8-28, 1990. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 54, Part 2, 337-357 (1993).
This article is a brief survey of selected topics in the homotopy theory of hyperplane arrangements. Let $${\mathcal A}$$ denote a finite collection of homogeneous hyperplanes in $$C^ n$$. The main object of interest in this theory is the homotopy type of the complement $$M=C^ n - \bigcup {\mathcal A}$$, especially in its dependence on the intersection lattice $$L({\mathcal A}) = \{X= \bigcap {\mathcal B} \mid {\mathcal B} \subseteq {\mathcal A}\}$$.
The authors begin with a discussion of the Orlik-Solomon presentation of the cohomology algebra $$H^*(M)$$ in terms of $$L({\mathcal A})$$. Next they focus on the higher homotopy groups, giving a precise description, due to Hattori, of the homotopy type of $$\pi_ i (M)$$, $$i \geq 2$$, when $${\mathcal A}$$ is generic. Supersolvable lattices and fiber-type arrangements are introduced, with primary emphasis on a formula of Kohno, Randell and the reviewer relating the lower central series of $$\pi_ 1 (M)$$ to $$H^*(M)$$. An approach to this formula using Sullivan’s theory of minimal models is described in some detail. Finally the authors present their own partial solution to the homotopy-type conjecture, that $$L({\mathcal A})$$ determines the homotopy type of $$M$$. In the case $$n=3$$ they give a graphical condition under which this conjecture holds. Most proofs are omitted. The paper closes with two nontrivial examples.
For the entire collection see [Zbl 0773.00023].

##### MSC:
 57R55 Differentiable structures in differential topology 14B05 Singularities in algebraic geometry 05B35 Combinatorial aspects of matroids and geometric lattices 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)