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The minimal model of the complement of an arrangement of hyperplanes


Author: Michael Falk
Journal: Trans. Amer. Math. Soc. 309 (1988), 543-556
MSC: Primary 32C40
DOI: https://doi.org/10.1090/S0002-9947-1988-0929668-7
MathSciNet review: 929668
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Abstract: In this paper the methods of rational homotopy theory are applied to a family of examples from singularity theory. Let $ {\mathbf{A}}$ be a finite collection of hyperplanes in $ {{\mathbf{C}}^l}$, and let $ M = {{\mathbf{C}}^l} - \bigcup\nolimits_{H \in {\mathbf{A}}} H $. We say $ {\mathbf{A}}$ is a rational $ K(\pi ,\,1)$ arrangement if the rational completion of $ M$ is aspherical. For these arrangements an identity (the LCS formula) is established relating the lower central series of $ {\pi _1}(M)$ to the cohomology of $ M$. This identity was established by group-theoretic means for the class of fiber-type arrangements in previous work. We reproduce this result by showing that the class of rational $ K(\pi ,\,1)$ arrangements contains all fiber-type arrangements. This class includes the reflection arrangements of types $ {A_l}$ and $ {B_l}$.

There is much interest in arrangements for which $ M$ is a $ K(\pi ,\,1)$ space. The methods developed here do not apply directly because $ M$ is rarely a nilpotent space. We give examples of $ K(\pi ,\,1)$ arrangements which are not rational $ K(\pi ,\,1)$ for which the LCS formula fails, and $ K(\pi ,\,1)$ arrangements which are not rational $ K(\pi ,\,1)$ where the LCS formula holds. It remains an open question whether rational $ K(\pi ,\,1)$ arrangements are necessarily $ K(\pi ,\,1)$.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0929668-7
Article copyright: © Copyright 1988 American Mathematical Society

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