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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 ΔW 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 ΔW.

Keywords
Milnor fibers, finite reflection groups, generalized braid groups, noncrossing partitions
Mathematical Subject Classification 2000
Primary: 20F55
Secondary: 52C35, 05E99
References
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