The diffeomorphism type of small hyperplane arrangements is combinatorially determined

It is known that there exist hyperplane arrangements with the same underlying matroid that admit non-homotopy equivalent complement manifolds. Here we show that, in any rank, complex central hyperplane arrangements with up to 7 hyperplanes and the same underlying matroid are isotopic. In particular,...

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Bibliographic Details
Published inAdvances in geometry Vol. 19; no. 1; pp. 89 - 100
Main Authors Gallet, Matteo, Saini, Elia
Format Journal Article
LanguageEnglish
Published Berlin De Gruyter 01.01.2019
Walter de Gruyter GmbH
Subjects
14N20
52B40
hyperplane arrangements
Hyperplanes
Isomorphism
Matroids
realization spaces
Set theory
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Summary:It is known that there exist hyperplane arrangements with the same underlying matroid that admit non-homotopy equivalent complement manifolds. Here we show that, in any rank, complex central hyperplane arrangements with up to 7 hyperplanes and the same underlying matroid are isotopic. In particular, the diffeomorphism type of the complement manifold and the Milnor fiber and fibration of these arrangements are combinatorially determined, that is, they depend only on the underlying matroid. To prove this, we associate to every such matroid a topological space, that we call the ; its connectedness, shown by means of symbolic computation, implies the desired result.
ISSN:1615-715X
1615-7168
DOI:10.1515/advgeom-2018-0015