On Formality and Combinatorial Formality for Hyperplane Arrangements

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

Authors

  • Tilman Möller
  • Paul Mücksch
  • Gerhard Röhrle

External Research Organisations

  • Ruhr-Universität Bochum
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Details

Original languageEnglish
Pages (from-to)73-90
Number of pages18
JournalDiscrete & computational geometry
Volume72
Issue number1
Early online date17 Feb 2023
Publication statusPublished - Jul 2024
Externally publishedYes

Abstract

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.

Keywords

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

ASJC Scopus subject areas

Cite this

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

Möller T, Mücksch P, Röhrle G. On Formality and Combinatorial Formality for Hyperplane Arrangements. Discrete & computational geometry. 2024 Jul;72(1):73-90. Epub 2023 Feb 17. doi: 10.1007/s00454-022-00479-5
Möller, Tilman ; Mücksch, Paul ; Röhrle, Gerhard. / On Formality and Combinatorial Formality for Hyperplane Arrangements. In: Discrete & computational geometry. 2024 ; Vol. 72, No. 1. pp. 73-90.
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