In this paper the methods of rational homotopy theory are applied to a family of examples from singularity theory. Let A be a finite collection of hyperplanes in C1, and let M = C1—HÎAH. We say A is a rational K(π, 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 π1(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(π, 1) arrangements contains all fiber-type arrangements. This class includes the reflection arrangements of types Aland Bl. There is much interest in arrangements for which M is a K(π, 1) space. The methods developed here do not apply directly because M is rarely a nilpotent space. We give examples of K(π, 1) arrangements which are not rational K(π, 1) for which the LCS formula fails, and K(π, 1) arrangements which are not rational K(π, 1) where the LCS formula holds. It remains an open question whether rational K(π, 1) arrangements are necessarily K(π, 1).
ASJC Scopus subject areas
- Applied Mathematics