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].

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].

Reviewer: M.J.Falk (Flagstaff)