Algebraic & Geometric Topology

Some analogs of Zariski's Theorem on nodal line arrangements

A D Raza Choudary, Alexandru Dimca, and Ştefan Papadima

Full-text: Open access

Abstract

For line arrangements in 2 with nice combinatorics (in particular, for those which are nodal away the line at infinity), we prove that the combinatorics contains the same information as the fundamental group together with the meridianal basis of the abelianization. We consider higher dimensional analogs of the above situation. For these analogs, we give purely combinatorial complete descriptions of the following topological invariants (over an arbitrary field): the twisted homology of the complement, with arbitrary rank one coefficients; the homology of the associated Milnor fiber and Alexander cover, including monodromy actions; the coinvariants of the first higher non-trivial homotopy group of the Alexander cover, with the induced monodromy action.

Article information

Source
Algebr. Geom. Topol., Volume 5, Number 2 (2005), 691-711.

Dates
Received: 18 October 2004
Revised: 12 May 2005
Accepted: 27 June 2005
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796426

Digital Object Identifier
doi:10.2140/agt.2005.5.691

Mathematical Reviews number (MathSciNet)
MR2153112

Zentralblatt MATH identifier
1081.32018

Subjects
Primary: 32S22: Relations with arrangements of hyperplanes [See also 52C35] 55N25: Homology with local coefficients, equivariant cohomology
Secondary: 14F35: Homotopy theory; fundamental groups [See also 14H30] 52C35: Arrangements of points, flats, hyperplanes [See also 32S22] 55Q52: Homotopy groups of special spaces

Keywords
hyperplane arrangement oriented topological type 1–marked group intersection lattice local system Milnor fiber Alexander cover

Citation

Choudary, A D Raza; Dimca, Alexandru; Papadima, Ştefan. Some analogs of Zariski's Theorem on nodal line arrangements. Algebr. Geom. Topol. 5 (2005), no. 2, 691--711. doi:10.2140/agt.2005.5.691. https://projecteuclid.org/euclid.agt/1513796426


Export citation

References

  • M. Bestvina and N. Brady: Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), 445–470.
  • D. Cohen, G. Denham and A. Suciu: Torsion in Milnor fiber homology, Algebraic and Geometric Topology 3 (2003), 511–535.
  • D. Cohen, A. Dimca and P. Orlik: Nonresonance conditions for arrangements, Annales de l'Insitut Fourier 53 (2003), 1883–1896.
  • D. Cohen and A. Suciu: On Milnor fibrations of arrangements, J. London Math. Soc. 51 (1995), 105–119.
  • P. Deligne: Le groupe fondamental du complément d'une courbe plane n'ayant que des points doubles ordinaires est abélien, Sém. Bourbaki 1979/80, LNM 842, pp.1–10, Springer-Verlag, Berlin 1981.
  • A. Dimca: Singularities and Topology of Hypersurfaces, Universitext, Springer-Verlag, 1992.
  • A. Dimca: Sheaves in Topology, Universitext, Springer-Verlag, 2004.
  • A. Dimca: Hyperplane arrangements, $M$–tame polynomials and twisted cohomology, Commutative Algebra, Singularities and Computer Algebra, Eds. J. Herzog, V. Vuletescu, NATO Science Series, Vol. 115, Kluwer (2003), 113–126.
  • A. Dimca and A. Némethi: Hypersurface complements, Alexander modules and monodromy, Proceedings of the 7th Workshop on Real and Complex Singularities, Sao Carlos, 2002, M. Ruas and T. Gaffney Eds., Contemp. Math. AMS 354 (2004), 19–43.
  • A. Dimca and S. Papadima: Hypersurface complements, Milnor fibers and higher homotopy groups of arrangements, Annals of Math. 158 (2003), 473–507.
  • A. Dimca and S. Papadima: Equivariant chain complexes, twisted homology and relative minimality of arrangements, Ann. Scient. Ec. Norm. Sup. 37 (2004), 449–467.
  • M. Falk: Homotopy types of line arrangements, Invent. Math. 111 (1993), 139–150.
  • K.–M. Fan: Direct product of free groups as the fundamental group of the complement of a union of lines, Michigan Math. J. 44 (1997), 283–291.
  • W. Fulton: On the fundamental group of the complement of a nodal curve, Annals of Math. 111 (1980), 407–409.
  • A. Hattori: Topology of $\C^n$ minus a finite number of affine hyperplanes in general position, J. Fac. Sci. Univ. Tokyo 22 (1975), 205–219.
  • J. A. Hillman: Alexander Ideals of Links, LNM 895, Springer-Verlag, 1981.
  • P. J. Hilton and U. Stammbach: A Course in Homological Algebra, Springer-Verlag, 1971.
  • M. Jambu and S. Papadima: A generalization of fiber–type arrangements and a new deformation method, Topology 37 (1998), 1135–1164.
  • T. Jiang and S.S.–T. Yau: Topological invariance of intersection lattices of arrangements in $\C \PP^2$, Bull. Amer. Math. Soc. 29 (1993), 88–93.
  • T. Jiang and S.S.–T. Yau: Diffeomorphic types of the complements of arrangements of hyperplanes, Compositio Math. 92 (1994), 133–155.
  • Y. Kawahara: The mixed Hodge structure on the fundamental group of a complement of hyperplanes, Topology and its Applications 118 (2002), 131–145.
  • Libgober, A.: Alexander modules of plane algebraic curves. Contemporary Mathematics, 20 (1983), 231–247
  • A. Libgober: Alexander invariants of plane algebraic curves, Proc. Symp. Pure Math. 40, Part 2, AMS (1983), 135–144.
  • A. Libgober: Homotopy groups of the complements to singular hypersurfaces, II, Annals of Math. 139 (1994), 117–144.
  • Libgober, A.: The topology of complements to hypersurfaces and nonvanishing of a twisted de Rham cohomology, AMS/IP Studies in Advanced Math. 5 (1997), 116–130
  • Libgober, A.: Eigenvalues for the monodromy of the Milnor fibers of arrangements. In: Libgober, A., Tibăr, M. (eds) Trends in Mathematics: Trends in Singularities. Birkhäuser, Basel (2002), 141–150
  • J. Milnor: Singular Points of Complex Hypersurfaces, Annals of Math. Studies 61, Princeton Univ. Press, 1968.
  • M. Oka and K. Sakamoto: Product theorem of the fundamental group of a reducible curve, J. Math. Soc. Japan 30 (1978), 599–602.
  • P. Orlik and L. Solomon: Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167–189.
  • P. Orlik and H. Terao: Arrangements of Hyperplanes, Grundlehren 300, Springer-Verlag, 1992.
  • P. Orlik and H. Terao: Arrangements and Hypergeometric Integrals, Math. Soc. of Japan Memoirs 9, Tokyo, 2001.
  • S. Papadima: Generalized ${\overline \mu}$–invariants for links and hyperplane arrangements, Proc. London Math. Soc. 84 (2002), 492–512.
  • S. Papadima and A. Suciu: Higher homotopy groups of complements of complex hyperplane arrangements, Advances in Math. 165 (2002), 71–100.
  • R. Randell: Lattice–isotopic arrangements are topologically isomorphic, Proc. Amer. Math. Soc. 107 (1989), 555–559.
  • R. Randell: Homotopy and group cohomology of arrangements, Topology Appl. 78 (1997), 201–213.
  • G. Rybnikov: On the fundamental group of the complement of a complex hyperplane arrangement, DIMACS Tech. Report 94-13 (1994), 33–50; available at \arxivmath.AG/9805056.
  • D. Siersma: The vanishing topology of non isolated singularities, in: D. Siersma et al.(eds) New Developments in Singularity Theory, Kluwer (2001), pp. 447–472.
  • J. R. Stallings: A finitely presented group whose $3$–dimensional integral homology is not finitely generated, Am. J. Math. 85 (1963), 541–543.
  • G. W. Whitehead: Elements of Homotopy Theory, Grad. Texts in Math. 61, Springer-Verlag, 1978.
  • O. Zariski: On the problem of existence of algebraic functions of two variables possessing a given branch curve, Amer. J. Math. 51 (1929), 305–328.