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Advances in Geometry

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The diffeomorphism type of small hyperplane arrangements is combinatorially determined

Matteo Gallet / Elia Saini
  • Corresponding author
  • Department of Mathematics, University of Fribourg, Chemin du Musée 23, 1700, Fribourg, Switzerland
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Published Online: 2018-10-25 | DOI: https://doi.org/10.1515/advgeom-2018-0015

Abstract

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 reduced realization space; its connectedness, shown by means of symbolic computation, implies the desired result.

Keywords: Matroids; hyperplane arrangements; realization spaces

MSC 2010: 14N20; 52B40

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About the article

Received: 2016-10-20

Accepted: 2017-01-27

Published Online: 2018-10-25


Communicated by: I. Coskun

Funding: The first-named author is supported by the Austrian Science Fund (FWF): W1214-N15/DK9 and P26607 - “Algebraic Methods in Kinematics: Motion Factorisation and Bond Theory”. The second-named author is supported by the Swiss National Science Foundation grant PP00P2_150552/1.


Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0015.

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