Multinets, parallel connections, and Milnor fibrations of arrangements.

*(English)*Zbl 1307.32024In the book [Singular points of complex hypersurfaces. Princeton, N.J.: Princeton University Press and the University of Tokyo Press (1968; Zbl 0184.48405)], J. W. Milnor associates to every homogeneous polynomial a fiber bundle with typical fiber called the Minor fiber. Milnor shows that if the polynomial has an isolated singularity at the origin, then the fiber is homotopic to a bouquet of circles. While considerable work has been done to study the fiber in other situations, the determination of the homology groups of the Milnor fiber has proven to be difficult.

In the present work, the authors study the fiber associated to a central arrangement of complex hyperplanes. D. C. Cohen and A. I. Suciu [J. Lond. Math. Soc., II. Ser. 51, No. 1, 105–119 (1995; Zbl 0814.32007)] provided an algorithm for determining the first homology group from a presentation of the fundamental group of the complement of the arrangement. In [Contemp. Math. 538, 309–318 (2011; Zbl 1309.14043)], R. Randell conjectured that the Milnor fiber had torsion-free homology, since the complement of the hyperplane arrangement has torsion-free homology. D. C. Cohen et al. [Algebr. Geom. Topol. 3, 511–535 (2003; Zbl 1030.32022)] showed that there exist multiarrangements (arrangements with unreduced equations) that have torsion-free homology complements, but whose Milnor fibers have torsion.

The present article shows for every prime \(p\) greater than \(2\) that there exists a hyperplane arrangement with Milnor fiber which has non-trivial \(p\)-torsion in homology. The authors begin with a multi-arrangement whose Milnor fiber is known to have torsion. They then take the underlying arrangement and form the parallel connection of the arrangement with several pencils of arrangements determined by the multiplicities in the multiarrangement. The resulting arrangement is completely reduced and has a Milnor fiber with torsion in certain degrees. The lowest degree where torsion is known to occur is degree seven.

Questions that remain are whether there exist arrangements whose Minor fibers have non-trivial torsion in degrees one through six and if the torsion is combinatorially determined.

In the present work, the authors study the fiber associated to a central arrangement of complex hyperplanes. D. C. Cohen and A. I. Suciu [J. Lond. Math. Soc., II. Ser. 51, No. 1, 105–119 (1995; Zbl 0814.32007)] provided an algorithm for determining the first homology group from a presentation of the fundamental group of the complement of the arrangement. In [Contemp. Math. 538, 309–318 (2011; Zbl 1309.14043)], R. Randell conjectured that the Milnor fiber had torsion-free homology, since the complement of the hyperplane arrangement has torsion-free homology. D. C. Cohen et al. [Algebr. Geom. Topol. 3, 511–535 (2003; Zbl 1030.32022)] showed that there exist multiarrangements (arrangements with unreduced equations) that have torsion-free homology complements, but whose Milnor fibers have torsion.

The present article shows for every prime \(p\) greater than \(2\) that there exists a hyperplane arrangement with Milnor fiber which has non-trivial \(p\)-torsion in homology. The authors begin with a multi-arrangement whose Milnor fiber is known to have torsion. They then take the underlying arrangement and form the parallel connection of the arrangement with several pencils of arrangements determined by the multiplicities in the multiarrangement. The resulting arrangement is completely reduced and has a Milnor fiber with torsion in certain degrees. The lowest degree where torsion is known to occur is degree seven.

Questions that remain are whether there exist arrangements whose Minor fibers have non-trivial torsion in degrees one through six and if the torsion is combinatorially determined.

Reviewer: Kristopher Williams (Crete)