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 be a finite collection of hyperplanes in , and let . We say is a *rational* arrangement if the rational completion of is aspherical. For these arrangements an identity (the LCS formula) is established relating the lower central series of to the cohomology of . 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 arrangements contains all fiber-type arrangements. This class includes the reflection arrangements of types and .

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

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DOI:
https://doi.org/10.1090/S0002-9947-1988-0929668-7

Article copyright:
© Copyright 1988
American Mathematical Society