Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Realizing homology boundary links
with arbitrary patterns


Author: Paul Bellis
Journal: Trans. Amer. Math. Soc. 350 (1998), 87-100
MSC (1991): Primary 57Q45, 57M07, 57M15
DOI: https://doi.org/10.1090/S0002-9947-98-01651-1
MathSciNet review: 1357391
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Homology boundary links have become an increasingly important class of links, largely due to their significance in the ongoing concordance classification of links. Tim Cochran and Jerome Levine defined an algebraic object called a pattern associated to a homology boundary link which can be used to study the deviance of a homology boundary link from being a boundary link. Since a pattern is a set of $m$ elements which normally generates the free group of rank $m$, any invariants which detect non-trivial patterns can be applied to the purely algebraic question of when such a set is a set of conjugates of a generating set for the free group. We will give a constructive geometric proof that all patterns are realized by some homology boundary link $L^n$ in $S^{n+2}$. We shall also prove an analogous existence theorem for calibrations of $\mathbb{E}$-links, a more general and less understood class of links tha homology boundary links.


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

  • 1. G. Bausmlaug, Groups with the Same Lower Central Sequence as a Relatively Free Group, Trans. Amer. Math. Soc. 142 (1969), 507-38. MR 39:6959
  • 2. S. Cappell and J. Shaneson, Link Cobordism, Comment. Math. Helv. 55 (1980), 20-49. MR 81j:57011
  • 3. T. Cochran, Link Concordance Invariants and Homotopy Theory, Invent. Math. 90 (1987), 635-45. MR 89f:57033
  • 4. -, Derivatives of Links: Milnor's Concordance Invariants and Massey's Products, #427, Memoirs of the Amer. Math. Soc. 84 (1990), Providence, RI. MR 91c:57005
  • 5. T. Cochran and J. Levine, Homology Boundary Links and the Andrews-Curtis Conjecture, Topology 30 (1991), 231-9. MR 92f:57011
  • 6. T. Cochran and K. Orr, Not All Links are Concordant to Boundary Links, Bull. Amer. Math. Soc. (N.S.) 23 (1990), 99-106. MR 91c:57012
  • 7. -, Not All Links are Concordant to Boundary Links, Ann. of Math. 138 (1993), 519-54. MR 95c:57042
  • 8. -, Homology Boundary Links and Blanchfield Forms: Concordance Classification and New Tangle-Theoretic Constructions, Topology 33 (1994), 397-427. MR 95f:57041
  • 9. R. De Meo, Cobordisms of Non-boundary Links, Ph.D. Dissertation, Princeton University, 1980.
  • 10. J. Hillman, Alexander Ideals of Links, Springer-Verlag Lecture Notes in Math. 895, Springer, Berlin, 1981. MR 84j:57004
  • 11. U. Kaiser, Homology Boundary Links and Fusion Constructions, Osaka J. Math. 29 (1992), 573-93. MR 93h:57038
  • 12. M. Kervaire, Les noeuds de dimensions supérieures, Bull. Soc. Math. France 93 (1965), 225-71. MR 32:6479
  • 13. K. Ko, Seifert Matrices and Boundary Link Cobordisms, Trans. Amer. Math. Soc. 299 (1987), 657-81. MR 88h:57018
  • 14. J. Levine, Knot Cobordism Groups in Codimension Two, Comment. Math. Helv. 44 (1969), 229-44. MR 39:7618
  • 15. -, Link Concordance and Algebraic Closure of Groups, Comment. Math. Helv. 64 (1989), 236-55. MR 91a:57016
  • 16. -, Link Concordance and Algebraic Closure II, Invent. Math. 96 (1989), 571-92. MR 91g:57007
  • 17. -, Link Invariants Via the Eta Invariant, Comment. Math. Helv. 69 (1994), 82-119. MR 95a:57009
  • 18. J. Levine, W. Mio, and K. Orr, Links With Vanishing Homotopy Invariants, Comm. Pure Appl. Math. 46 (1993), 213-20. MR 94e:57036
  • 19. W. Mio, On Boundary Link Cobordism, Math. Proc. Cambridge Philos. Soc. 101 (1987), 259-66. MR 88e:57023
  • 20. N. Smythe, Boundary Links, Topology Seminar: Wisconsin, 1965 (ed. R. H. Bing), Annals of Math. Studies 60, Princeton Univ. Press, Princeton, NJ, pp. 59-72.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 57Q45, 57M07, 57M15

Retrieve articles in all journals with MSC (1991): 57Q45, 57M07, 57M15


Additional Information

Paul Bellis
Affiliation: Department of Mathematics, Rice University, P. O. Box 1892, Houston, Texas 77251-1892
Address at time of publication: 7932 Butterfield Dr., Elkridge, Maryland 21075
Email: apbellis@erols.com

DOI: https://doi.org/10.1090/S0002-9947-98-01651-1
Received by editor(s): May 16, 1995
Received by editor(s) in revised form: October 30, 1995
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society