## Details

Original language | English |
---|---|

Pages (from-to) | 73-90 |

Number of pages | 18 |

Journal | Discrete & computational geometry |

Volume | 72 |

Issue number | 1 |

Early online date | 17 Feb 2023 |

Publication status | Published - Jul 2024 |

Externally published | Yes |

## Abstract

## Keywords

- Combinatorial formality, Factored arrangements, Formality, Free arrangements, Hyperplane arrangements, K(π, 1)-Arrangements, k-Formality

## ASJC Scopus subject areas

- Mathematics(all)
**Theoretical Computer Science**- Mathematics(all)
**Discrete Mathematics and Combinatorics**- Mathematics(all)
**Geometry and Topology**- Computer Science(all)
**Computational Theory and Mathematics**

## Cite this

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- RIS

**On Formality and Combinatorial Formality for Hyperplane Arrangements.**/ Möller, Tilman; Mücksch, Paul; Röhrle, Gerhard.

In: Discrete & computational geometry, Vol. 72, No. 1, 07.2024, p. 73-90.

Research output: Contribution to journal › Article › Research › peer review

*Discrete & computational geometry*, vol. 72, no. 1, pp. 73-90. https://doi.org/10.1007/s00454-022-00479-5

*Discrete & computational geometry*,

*72*(1), 73-90. https://doi.org/10.1007/s00454-022-00479-5

}

TY - JOUR

T1 - On Formality and Combinatorial Formality for Hyperplane Arrangements

AU - Möller, Tilman

AU - Mücksch, Paul

AU - Röhrle, Gerhard

N1 - Publisher Copyright: © 2023, The Author(s).

PY - 2024/7

Y1 - 2024/7

N2 - A hyperplane arrangement is called formal provided all linear dependencies among the defining forms of the hyperplanes are generated by ones corresponding to intersections of codimension two. The significance of this notion stems from the fact that complex arrangements with aspherical complements are formal. The aim of this note is twofold. While work of Yuzvinsky shows that formality is not combinatorial, in our first main theorem we prove that the combinatorial property of factoredness of arrangements does entail formality. Our second main theorem shows that formality is hereditary, i.e., is passed to restrictions. This is rather counter-intuitive, as in contrast the known sufficient conditions for formality, i.e., asphericity, freeness and factoredness (owed to our first theorem), are not hereditary themselves. We also demonstrate that the stronger property of k-formality, due to Brandt and Terao, is not hereditary.

AB - A hyperplane arrangement is called formal provided all linear dependencies among the defining forms of the hyperplanes are generated by ones corresponding to intersections of codimension two. The significance of this notion stems from the fact that complex arrangements with aspherical complements are formal. The aim of this note is twofold. While work of Yuzvinsky shows that formality is not combinatorial, in our first main theorem we prove that the combinatorial property of factoredness of arrangements does entail formality. Our second main theorem shows that formality is hereditary, i.e., is passed to restrictions. This is rather counter-intuitive, as in contrast the known sufficient conditions for formality, i.e., asphericity, freeness and factoredness (owed to our first theorem), are not hereditary themselves. We also demonstrate that the stronger property of k-formality, due to Brandt and Terao, is not hereditary.

KW - Combinatorial formality

KW - Factored arrangements

KW - Formality

KW - Free arrangements

KW - Hyperplane arrangements

KW - K(π, 1)-Arrangements

KW - k-Formality

UR - http://www.scopus.com/inward/record.url?scp=85148370940&partnerID=8YFLogxK

U2 - 10.1007/s00454-022-00479-5

DO - 10.1007/s00454-022-00479-5

M3 - Article

VL - 72

SP - 73

EP - 90

JO - Discrete & computational geometry

JF - Discrete & computational geometry

SN - 0179-5376

IS - 1

ER -