### Abstract

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 C^{1}, and let M = C^{1}—_{H} _{ÎA}H. 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 π_{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(π, 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(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).

Original language | English (US) |
---|---|

Pages (from-to) | 543-556 |

Number of pages | 14 |

Journal | Transactions of the American Mathematical Society |

Volume | 309 |

Issue number | 2 |

DOIs | |

State | Published - 1988 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**The minimal model of the complement of an arrangement of hyperplanes.** / Falk, Michael J.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The minimal model of the complement of an arrangement of hyperplanes

AU - Falk, Michael J

PY - 1988

Y1 - 1988

N2 - 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—H ÎAH. 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).

AB - 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—H ÎAH. 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).

UR - http://www.scopus.com/inward/record.url?scp=0001144529&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001144529&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-1988-0929668-7

DO - 10.1090/S0002-9947-1988-0929668-7

M3 - Article

AN - SCOPUS:0001144529

VL - 309

SP - 543

EP - 556

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -