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Zbl 0828.57018
Jiang, Tan; Yau, Stephen S.-T.
Diffeomorphic types of the complements of arrangements of hyperplanes. (English)
[J] Compos. Math. 92, No.2, 133-155 (1994). ISSN 0010-437X; ISSN 1570-5846/e

A fundamental open problem in the theory of complex hyperplane arrangements has been the conjecture that the homotopy or topological type of the complement of a finite collection of hyperplanes is a function only of the underlying matroid or intersection lattice [{\it P. Orlik} and {\it H. Terao}, Arrangements of hyperplanes, Grundlehren Math. Wiss. 300 (1992; Zbl 0757.55001)]. This problem has now been resolved in the negative in unpublished work of {\it Grigory Rybnikov} [On the fundamental group of a complex hyperplane arrangement (preprint, 1993)]. In the paper under review, the authors provide a sufficient condition for the conjecture to hold. Given an arrangement ${\cal A}$ of lines in $CP^2$, construct the graph $G({\cal A})$ whose vertices are the points of multiplicity greater than two, with two vertices adjacent when there is a line of ${\cal A}$ containing them. The star of a vertex is the set of lines containing it (which may contain several collinear edges). An arrangement is called ``nice'' if there is a set of pairwise disjoint stars in $G({\cal A})$ whose complement is a forest. In this case it is shown that for any arrangement ${\cal A}'$ with the same intersection lattice as ${\cal A}$, the arrangements along the segment from ${\cal A}$ to ${\cal A}'$ have constant intersection lattice. In other words, the realization space of the matroid of ${\cal A}$ is convex. Then the lattice isotopy theorem of {\it R. Randell} [Proc. Am. Math. Soc. 107, 555-559 (1989; Zbl 0681.57016)] implies that ${\cal A}$ and ${\cal A}'$ have diffeomorphic complements.
[M.J.Falk (Madison)]

MSC 2000:: *57R19 Algebraic topology on manifolds
57M05 Fundamental group, etc.
52C35 Arrangements of points, flats, hyperplanes

Keywords: homotopy type of the complement; lines in $CP\sp 2$; complex hyperplane arrangements; intersection lattice

Citations: Zbl 0757.55001; Zbl 0681.57016

Cited in: Zbl 1243.14047 Zbl 1115.52010

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