Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Morse theory, Milnor fibers and minimality of hyperplane arrangements

Author: Richard Randell
Journal: Proc. Amer. Math. Soc. 130 (2002), 2737-2743
MSC (2000): Primary 52C35, 55Q52; Secondary 14N20, 32S22
Published electronically: February 4, 2002
MathSciNet review: 1900880
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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-MacLane spaces.

References [Enhancements On Off] (What's this?)

  • 1. Milnor, J., Singular points of complex hypersurfaces, Annals of Math. Studies 61, Princeton University Press, 1968. MR 39:969
  • 2. Lê, D. T., Calcul du nombre de cycles évanouissants d'une hypersurface complexe, Ann. Inst. Fourier, Grenoble 23, 4 (1973), 261-270. MR 48:8838
  • 3. Papadima, S. and Suciu, A., Higher homotopy groups of complements of complex hyperplane arrangements, preprint, arXiv:math.AT/0002251
  • 4. Orlik, P., and Terao, H., Arrangements of Hyperplanes, Grundlehren der mathematischen Wissenschaften 300, Springer Verlag, 1992. MR 94e:52014
  • 5. -, Arrangements and Milnor fibers, Math. Ann. 301, (1995), 211-235. MR 96f:52014
  • 6. Dimca, A., Hypersurface complements, Milnor fibers and minimality of arrangements, preprint, arXiv:math.AG/0011222.
  • 7. Randell, R., Homotopy and group cohomology of arrangements, Topology and its Applications 78, (1997), 201-213. MR 98f:52014
  • 8. Artal Bartolo, E., Combinatorics and topology of line arrangements in the complex projective plane, Proc. Amer. Math. Soc., 121, (1994), 385-390. MR 94h:14020
  • 9. Arvola, B., Arrangements and cohomology of groups, preprint, 1992.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 52C35, 55Q52, 14N20, 32S22

Retrieve articles in all journals with MSC (2000): 52C35, 55Q52, 14N20, 32S22

Additional Information

Richard Randell
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242

Keywords: Hyperplane arrangement, Milnor fiber, Morse theory
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
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society