Torsion in Milnor fiber homology

2003/02/28 by Daniel C. Cohen, Daniel C Cohen, Graham Denham +2
Mathematics · #Advanced Combinatorial Mathematics #Alexander polynomial #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Homology (biology) #Hyperplane #Hypersurface #Seifert surface #Torsion (gastropod) #math.AG #math.GT #msc:14J70 #msc:32S22 #msc:32S55 #msc:55N25

paper · pdf · doi:10.2140/agt.2003.3.511

published as Algebr. Geom. Topol. 3 (2003) 511-535 · Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-16.abs.html

openalex publication_date 2003/06/15 · arxiv created 2003/07/10 · arxiv updated 2014/09/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

In a recent paper, Dimca and Nmethi pose the problem of finding a homogeneous polynomial f such that the homology of the complement of the hypersurface defined by f is torsion-free, but the homology of the Milnor fiber of f has torsion. We prove that this is indeed possible, and show by construction that, for each prime p, there is a polynomial with p-torsion in the homology of the Milnor fiber. The techniques make use of properties of characteristic varieties of hyperplane arrangements.

Citations

• Arrangements of Hyperplanes
• HALL INVARIANTS, HOMOLOGY OF SUBGROUPS, AND CHARACTERISTIC VARIETIES
• Homology of iterated semidirect products of free groups
• The braid monodromy of plane algebraic curves and hyperplane arrangements
• Characteristic varieties of arrangements