The minimal model of the complement of an arrangement of hyperplanes

Falk, Michael

doi:10.1090/S0002-9947-1988-0929668-7

The minimal model of the complement of an arrangement of hyperplanes
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Trans. Amer. Math. Soc. 309 (1988), 543-556 Request permission

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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Poincaré series of the Malcev completion of generalized pure braid groups

Rational

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formula

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