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Transactions of the American Mathematical Society

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Total linking number modules


Author: Oziride Manzoli Neto
Journal: Trans. Amer. Math. Soc. 307 (1988), 503-533
MSC: Primary 57Q45
DOI: https://doi.org/10.1090/S0002-9947-1988-0940215-6
MathSciNet review: 940215
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Abstract: Given a codimension two link $ L$ in a sphere $ {S^k}$ with complement $ X = {S^k} - L$, the total linking number covering of $ L$ is the covering $ \hat X \to X$ associated to the surjection $ {\pi _1}(X) \to Z$ defined by sending the meridians to $ 1$. The homology $ {H_{\ast}}(\hat X)$ define weaker invariants than the homology of the universal abelian covering of $ L$.

The groups $ {H_i}(\hat X)$ are modules over $ Z\left[ {t,\,{t^{ - 1}}} \right]$ and this work gives an algebraic characterization of these modules for $ k \geqslant 4$ except for the pseudo null part of $ {H_1}(\hat X)$.


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

  • 1. [B.1) N. Bourbaki, Éléments de mathématiques, XXVII, Algèbre Commutative, Hermann, Paris, 1968.
  • [B.2] J. L. Bayley, Alexander invariants of links, Ph.D. Thesis, Univ. of British Columbia, 1977.
  • [C.E.] S. Eilemberg and H. Cartan, Homological algebra, Princeton Univ. Press, Princeton, N.J., 1956. MR 0077480 (17:1040e)
  • [G] M. A. Gutiérrez, Links of codimension two, Revista Colombiana de Matemáticas, 1970.
  • [H.1] J. A. Hillman, A link with Alexander module free, which is not a homotopy boundary link, J. Pure Appl. Algebra 20 (1981), 1-5. MR 596149 (82d:57001)
  • [H.2] -, Alexander ideals of links, Lecture Notes in Math., vol. 895, Springer-Verlag, Berlin and New York, 1981. MR 653808 (84j:57004)
  • [K] M. A. Kervaire, On higher dimensional knots, Symposium Marston Morse, Princeton, N.J., 1965. MR 0178475 (31:2732)
  • [L.1] J. Levine, Knot modules. I, Trans. Amer. Math. Soc. 229 (1977), 1-50. MR 0461518 (57:1503)
  • [L.2] -, The module of a $ 2$-component link (preprint).
  • [L.3] -, Unknotting spheres in codimension two, Topology 4 (1965), 9-16. MR 0179803 (31:4045)
  • [L.4] -, A survey of results concerning the embedding of polyhedra on manifolds in Euclidean spaces (unpublished).
  • [M.1] H. Matsumura, Commutative algebra, Benjamin/Cummings, 1980. MR 575344 (82i:13003)
  • [M.2] J. Milnor, A duality theorem for Reidmeister torsion, Ann. of Math. 76 (1962), MR 0141115 (25:4526)
  • [N] L. P. Neuwirth, Knot groups, Ann. of Math. Studies, no. 56, Princeton Univ. Press, 1965.
  • [R] D. Rolfsen, Knots and links, Publish or Perish, 1976. MR 0515288 (58:24236)
  • [S.1] C. Seshadri, Triviality of vector bundles over the affine space $ {K^2}$, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 456-458. MR 0102527 (21:1318)
  • [S.2] Y. Shinohara and D. W. Summers, Homotopy Invariants of cyclic coverings with applications to links, Trans. Amer. Math. Soc. 163 (1972), 101-120. MR 0284999 (44:2223)
  • [T] H. Trotter, On $ S$-equivalence of Seifert matrices, Invent. Math. 20 (1973), 173-207. MR 0645546 (58:31100)
  • [V.1] W. V. Vasconcelos, On local and stable cancelation, An. Acad. Brasil. de Ciênc. 37 MR 0214591 (35:5440)
  • [V.2] E. H. van Kampen, Komplexe in euklidischen Raümen, Abh. Math. Sem. Univ. Hamburg 9 (1932), 72-78.

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DOI: https://doi.org/10.1090/S0002-9947-1988-0940215-6
Article copyright: © Copyright 1988 American Mathematical Society

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