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Transactions of the American Mathematical Society

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Alexander invariants
of complex hyperplane arrangements

Authors: Daniel C. Cohen and Alexander I. Suciu
Journal: Trans. Amer. Math. Soc. 351 (1999), 4043-4067
MSC (1991): Primary 14H30, 52B30, 57M05; Secondary 20F14, 20F36
Published electronically: April 27, 1999
MathSciNet review: 1475679
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $\mathcal{A}$ be an arrangement of $n$ complex hyperplanes. The fundamental group of the complement of $\mathcal{A}$ is determined by a braid monodromy homomorphism, $\alpha:F_{s}\to P_{n}$. Using the Gassner representation of the pure braid group, we find an explicit presentation for the Alexander invariant of $\mathcal{A}$. From this presentation, we obtain combinatorial lower bounds for the ranks of the Chen groups of $\mathcal{A}$. We also provide a combinatorial criterion for when these lower bounds are attained.

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  • 1. W. Arvola, Arrangements and cohomology of groups, preprint.
  • 2. T. Becker, V. Weispfenning, Gröbner bases, Grad. Texts in Math., vol. 141, Springer-Verlag, New York-Berlin-Heidelberg, 1993.MR 95e:13018
  • 3. J. Birman, Braids, links and mapping class groups, Annals of Math. Studies, vol. 82, Princeton Univ. Press, Princeton, NJ, 1975.MR 51:11477
  • 4. K. T. Chen, Integration in free groups, Annals of Math. 54 (1951), 147-162. MR 13:105c
  • 5. D. Cohen, A. Suciu, The Chen groups of the pure braid group, In: Proceedings of the \v{C}ech Centennial Homotopy Theory Conference, Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 45-64. MR 96c:20055
  • 6. -, Homology of iterated semidirect products of free groups, J. Pure Appl. Algebra 126 (1998), 87-120. CMP 98:07
  • 7. -, The braid monodromy of plane algebraic curves and hyperplane arrangements, Comment. Math. Helvetici 72 (1997), 285-315. MR 98f:52012
  • 8. R. Cordovil, J. Fachada, Braid monodromy groups of wiring diagrams, Boll. Unione Mat. Ital. 9 (1995), 399-416.MR 96e:20057
  • 9. D. Cox, J. Little, D. O'Shea, Ideals, varieties, and algorithms, 2nd ed., Undergrad. Texts Math., Springer-Verlag, New York-Berlin-Heidelberg, 1997. MR 93j:13031
  • 10. R. H. Crowell, Torsion in link modules, J. Math. Mech. 14 (1965), 289-298. MR 30:4807
  • 11. -, The derived module of a homomorphism, Adv. in Math. 6 (1971), 210-238. MR 43:2055
  • 12. M. Falk, The minimal model of the complement of an arrangement of hyperplanes, Trans. Amer. Math. Soc. 309 (1988), 543-556. MR 89d:32024
  • 13. -, The cohomology and fundamental group of a hyperplane complement, In: Singularities, Contemp. Math., vol. 90, Amer. Math. Soc., Providence, RI, 1989, pp. 55-72. MR 90h:32026
  • 14. -, Arrangements and cohomology, Ann. Combin. 1 (1997), 135-157. CMP 98:14
  • 15. M. Falk, R. Randell, The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985), 77-88. MR 87c:32015b
  • 16. -, On the homotopy theory of arrangements, In: Complex Analytic Singularities, Adv. Stud. Pure Math., vol. 8, North Holland, Amsterdam, 1987, pp. 101-124. MR 88f:32045
  • 17. R. Fox, Free differential calculus I, Annals of Math. 57 (1953), 547-560; II, 59 (1954), 196-210; III, 64 (1956), 407-419. MR 14:843d; MR 15:931e; MR 20:2374
  • 18. M. Goresky, R. MacPherson, Stratified Morse theory, Ergeb. Math. Grenzgeb., vol. 14, Springer-Verlag, New York-Berlin-Heidelberg, 1988. MR 84k:58017
  • 19. H. Hamm, Lê D. T., Un théorème de Zariski du type de Lefschetz, Ann. Sci. École Norm. Sup. 6 (1973), 317-366. MR 53:5582
  • 20. E. Hironaka, Alexander stratifications of character varieties, Ann. Inst. Fourier (Grenoble) 47 (1997), 555-583. MR 98e:14020
  • 21. A. Libgober, On the homotopy type of the complement to plane algebraic curves, J. Reine Angew. Math. 367 (1986), 103-114. MR 87j:14044
  • 22. -, Abelian branched covers of projective plane, In: Singularity Theory, London Mathematical Society Lecture Note Series (J. W. Bruce, D. Mond, eds.), Cambridge Univ. Press, 1999, to appear.
  • 23. W. Massey, Completion of link modules, Duke Math. J. 47 (1980), 399-420. MR 81g:57004
  • 24. W. Massey, L. Traldi, On a conjecture of K. Murasugi, Pacific J. Math. 124 (1986), 193-213. MR 87k:57008
  • 25. S. Moran, The mathematical theory of knots and braids, North Holland Math. Stud., vol. 82, North Holland, Amsterdam, 1983. MR 85i:57001
  • 26. K. Murasugi, On Milnor's invariants for links. II. The Chen groups, Trans. Amer. Math. Soc. 148 (1970), 41-61. MR 41:4519
  • 27. P. Orlik, H. Terao, Arrangements of hyperplanes, Grundlehren Math. Wiss., vol. 300, Springer-Verlag, New York-Berlin-Heidelberg, 1992. MR 94e:52014
  • 28. R. Randell, Homotopy and group cohomology of arrangements, Top. and Appl. 20 (1996), 1-13. MR 98f:52014
  • 29. G. Rybnikov, On the fundamental group of the complement of a complex hyperplane arrangement, DIMACS Tech. Report 94-13 (1994), pp. 4043-4067; math. AG/9805056.
  • 30. B. Shelton, S. Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements, J. London Math. Soc. 56 (1997), 477-490. CMP 98:09
  • 31. L. Traldi, The determinantal ideals of link modules. I, Pacific J. Math. 101 (1982), 215-222.MR 84h:57004
  • 32. O. Zariski, P. Samuel, Commutative algebra, vols. 1 and 2, reprint of the 1958-1960 edition, Springer-Verlag, New York-Berlin-Heidelberg, 1979. MR 52:5641; MR 52:10706

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Additional Information

Daniel C. Cohen
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803

Alexander I. Suciu
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115

Keywords: Arrangement, braid monodromy, Alexander invariant, Chen groups
Received by editor(s): March 24, 1997
Received by editor(s) in revised form: September 9, 1997
Published electronically: April 27, 1999
Additional Notes: The first author was partially supported by grant LEQSF(1996-99)-RD-A-04 from the Louisiana Board of Regents and by a grant from the Louisiana State University Council on Research.
The second author was partially supported by N.S.F. grant DMS–9504833, and an RSDF grant from Northeastern University.
Article copyright: © Copyright 1999 American Mathematical Society

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