Volume 18, issue 7 (2018)

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Noncrossing partitions and Milnor fibers

Thomas Brady, Michael J Falk and Colum Watt

Algebraic & Geometric Topology 18 (2018) 3821–3838
Abstract

For a finite real reflection group $W$ we use noncrossing partitions of type $W$ to construct finite cell complexes with the homotopy type of the Milnor fiber of the associated $W$–discriminant ${\Delta }_{W}$ and that of the Milnor fiber of the defining polynomial of the associated reflection arrangement. These complexes support natural cyclic group actions realizing the geometric monodromy. Using the shellability of the noncrossing partition lattice, this cell complex yields a chain complex of homology groups computing the integral homology of the Milnor fiber of ${\Delta }_{W}$.

Keywords
Milnor fibers, finite reflection groups, generalized braid groups, noncrossing partitions
Mathematical Subject Classification 2000
Primary: 20F55
Secondary: 52C35, 05E99
Publication
Received: 16 June 2017
Revised: 12 June 2018
Accepted: 21 June 2018
Published: 11 December 2018
Authors
 Thomas Brady School of Mathematical Sciences Dublin City University Dublin Ireland Michael J Falk Department Mathematics and Statistics Northern Arizona University Flagstaff, AZ United States Colum Watt School of Mathematical Sciences Dublin Institute of Technology Dublin Ireland