Morse theory, Milnor fibers and minimality of hyperplane arrangements
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- by Richard Randell PDF
- Proc. Amer. Math. Soc. 130 (2002), 2737-2743 Request permission
Abstract:
Through the study of Morse theory on the associated Milnor fiber, we show that complex hyperplane arrangement complements are minimal. That is, the complement of any complex hyperplane arrangement has the homotopy type of a CW-complex in which the number of $p$-cells equals the $p$-th betti number. Combining this result with recent work of Papadima and Suciu, one obtains a characterization of when arrangement complements are Eilenberg-Mac Lane spaces.References
- John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
- Lê Dũng Tráng, Calcul du nombre de cycles évanouissants d’une hypersurface complexe, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 4, 261–270 (French, with English summary). MR 330501
- Papadima, S. and Suciu, A., Higher homotopy groups of complements of complex hyperplane arrangements, preprint, arXiv:math.AT/0002251
- Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR 1217488, DOI 10.1007/978-3-662-02772-1
- Peter Orlik and Hiroaki Terao, Arrangements and Milnor fibers, Math. Ann. 301 (1995), no. 2, 211–235. MR 1314585, DOI 10.1007/BF01446627
- Dimca, A., Hypersurface complements, Milnor fibers and minimality of arrangements, preprint, arXiv:math.AG/0011222.
- Richard Randell, Homotopy and group cohomology of arrangements, Topology Appl. 78 (1997), no. 3, 201–213. MR 1454600, DOI 10.1016/S0166-8641(96)00123-X
- Enrique Artal-Bartolo, Combinatorics and topology of line arrangements in the complex projective plane, Proc. Amer. Math. Soc. 121 (1994), no. 2, 385–390. MR 1189536, DOI 10.1090/S0002-9939-1994-1189536-3
- Arvola, B., Arrangements and cohomology of groups, preprint, 1992.
Additional Information
- Richard Randell
- Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
- Email: randell@math.uiowa.edu
- Received by editor(s): November 29, 2000
- Received by editor(s) in revised form: April 16, 2001
- Published electronically: February 4, 2002
- Communicated by: Ronald A. Fintushel
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 2737-2743
- MSC (2000): Primary 52C35, 55Q52; Secondary 14N20, 32S22
- DOI: https://doi.org/10.1090/S0002-9939-02-06412-2
- MathSciNet review: 1900880