Algebraic & Geometric Topology

Noncrossing partitions and Milnor fibers

Thomas Brady, Michael J Falk, and Colum Watt

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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 Δ 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 Δ W .

Article information

Algebr. Geom. Topol., Volume 18, Number 7 (2018), 3821-3838.

Received: 16 June 2017
Revised: 12 June 2018
Accepted: 21 June 2018
First available in Project Euclid: 18 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]
Secondary: 52C35: Arrangements of points, flats, hyperplanes [See also 32S22] 05E99: None of the above, but in this section

Milnor fibers finite reflection groups generalized braid groups noncrossing partitions


Brady, Thomas; Falk, Michael J; Watt, Colum. Noncrossing partitions and Milnor fibers. Algebr. Geom. Topol. 18 (2018), no. 7, 3821--3838. doi:10.2140/agt.2018.18.3821.

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  • D Allcock, T Basak, Geometric generators for braid-like groups, Geom. Topol. 20 (2016) 747–778
  • D Armstrong, Generalized noncrossing partitions and combinatorics of Coxeter groups, Mem. Amer. Math. Soc. 949, Amer. Math. Soc., Providence, RI (2009)
  • C A Athanasiadis, T Brady, C Watt, Shellability of noncrossing partition lattices, Proc. Amer. Math. Soc. 135 (2007) 939–949
  • D Bessis, The dual braid monoid, Ann. Sci. École Norm. Sup. 36 (2003) 647–683
  • A Björner, Shellable and Cohen–Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980) 159–183
  • N Bourbaki, Lie groups and Lie algebras, Chapters $7$–$9$, Springer (2005)
  • T Brady, A partial order on the symmetric group and new $K(\pi,1)$'s for the braid groups, Adv. Math. 161 (2001) 20–40
  • T Brady, C Watt, $K(\pi,1)$'s for Artin groups of finite type, Geom. Dedicata 94 (2002) 225–250
  • T Brady, C Watt, Non-crossing partition lattices in finite real reflection groups, Trans. Amer. Math. Soc. 360 (2008) 1983–2005
  • E Brieskorn, Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe, Invent. Math. 12 (1971) 57–61
  • E Brieskorn, Sur les groupes de tresses (d'après V I Arnol'd), from “Séminaire Bourbaki, 1971/1972”, Lecture Notes in Math. 317, Springer (1973) Exposé 401, 21–44
  • F Callegaro, The homology of the Milnor fiber for classical braid groups, Algebr. Geom. Topol. 6 (2006) 1903–1923
  • F Callegaro, M Salvetti, Integral cohomology of the Milnor fibre of the discriminant bundle associated with a finite Coxeter group, C. R. Math. Acad. Sci. Paris 339 (2004) 573–578
  • C Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955) 778–782
  • C De Concini, M Salvetti, Cohomology of Artin groups, Math. Res. Lett. 3 (1996) 293–297
  • C De Concini, M Salvetti, F Stumbo, The top-cohomology of Artin groups with coefficients in rank-$1$ local systems over ${\mathbb Z}$, Topology Appl. 78 (1997) 5–20
  • P Dehornoy, F Digne, E Godelle, D Krammer, J Michel, Foundations of Garside theory, EMS Tracts in Mathematics 22, Eur. Math. Soc., Zürich (2015)
  • P Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972) 273–302
  • A Dimca, G Lehrer, Cohomology of the Milnor fibre of a hyperplane arrangement with symmetry, from “Configuration spaces” (F Callegaro, F Cohen, C De Concini, E M Feichtner, G Gaiffi, M Salvetti, editors), Springer INdAM Ser. 14, Springer (2016) 233–274
  • \relax È V Frenkel', Cohomology of the commutator subgroup of the braid group, Funktsional. Anal. i Prilozhen. 22 (1988) 91–92 In Russian; translated in
  • A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)
  • J E Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29, Cambridge Univ. Press (1990)
  • A Kenny, Geometrically constructed bases for homology of non-crossing partition lattices, Electron. J. Combin. 16 (2009) art. id. 48, 8 pages
  • D Kozlov, Combinatorial algebraic topology, Algorithms and Computation in Mathematics 21, Springer (2008)
  • J Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies 61, Princeton Univ. Press (1968)
  • J Milnor, P Orlik, Isolated singularities defined by weighted homogeneous polynomials, Topology 9 (1970) 385–393
  • A D Măcinic, \relax \commaaccentS Papadima, On the monodromy action on Milnor fibers of graphic arrangements, Topology Appl. 156 (2009) 761–774
  • J R Munkres, Elements of algebraic topology, Addison-Wesley (1984)
  • M Salvetti, The homotopy type of Artin groups, Math. Res. Lett. 1 (1994) 565–577
  • S Settepanella, A stability-like theorem for cohomology of pure braid groups of the series $A$, $B$ and $D$, Topology Appl. 139 (2004) 37–47
  • S Settepanella, Cohomology of pure braid groups of exceptional cases, Topology Appl. 156 (2009) 1008–1012