Submanifolds of codimension two and homology equivalent manifolds

Sylvain E. Cappell; Julius L. Shaneson

Annales de l'institut Fourier (1973)

  • Volume: 23, Issue: 2, page 19-30
  • ISSN: 0373-0956

Abstract

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In this paper new methods of studying codimension two embeddings of manifolds are outlined. Results are stated on geometric periodicity of knot cobordism. The group of local knots of a manifold in a 2-plane bundle is introduced and computed, and applied to C o -close embeddings. General codimension two splitting theorems are discussed, with applications to equivariant knots and knot cobordism. A general existence theorem for P.L. (non-locally flat) embeddings is also given.The methods involve some new functors in Hermitian K -theory, denoted Γ n ( I ) . Some of the results are stated in terms of these functors and a “Kunneth formula” for Γ n ( I × Z ) is indicated.

How to cite

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Cappell, Sylvain E., and Shaneson, Julius L.. "Submanifolds of codimension two and homology equivalent manifolds." Annales de l'institut Fourier 23.2 (1973): 19-30. <http://eudml.org/doc/74124>.

@article{Cappell1973,
abstract = {In this paper new methods of studying codimension two embeddings of manifolds are outlined. Results are stated on geometric periodicity of knot cobordism. The group of local knots of a manifold in a 2-plane bundle is introduced and computed, and applied to $C^o$-close embeddings. General codimension two splitting theorems are discussed, with applications to equivariant knots and knot cobordism. A general existence theorem for P.L. (non-locally flat) embeddings is also given.The methods involve some new functors in Hermitian $K$-theory, denoted $\Gamma _n(\{\bf I\})$. Some of the results are stated in terms of these functors and a “Kunneth formula” for $\Gamma _n(\{\bf I\}\times Z)$ is indicated.},
author = {Cappell, Sylvain E., Shaneson, Julius L.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {2},
pages = {19-30},
publisher = {Association des Annales de l'Institut Fourier},
title = {Submanifolds of codimension two and homology equivalent manifolds},
url = {http://eudml.org/doc/74124},
volume = {23},
year = {1973},
}

TY - JOUR
AU - Cappell, Sylvain E.
AU - Shaneson, Julius L.
TI - Submanifolds of codimension two and homology equivalent manifolds
JO - Annales de l'institut Fourier
PY - 1973
PB - Association des Annales de l'Institut Fourier
VL - 23
IS - 2
SP - 19
EP - 30
AB - In this paper new methods of studying codimension two embeddings of manifolds are outlined. Results are stated on geometric periodicity of knot cobordism. The group of local knots of a manifold in a 2-plane bundle is introduced and computed, and applied to $C^o$-close embeddings. General codimension two splitting theorems are discussed, with applications to equivariant knots and knot cobordism. A general existence theorem for P.L. (non-locally flat) embeddings is also given.The methods involve some new functors in Hermitian $K$-theory, denoted $\Gamma _n({\bf I})$. Some of the results are stated in terms of these functors and a “Kunneth formula” for $\Gamma _n({\bf I}\times Z)$ is indicated.
LA - eng
UR - http://eudml.org/doc/74124
ER -

References

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  1. [1] W. BROWDER, Embedding smooth manifolds, In «Proceedings of the International Congress of Mathematicians (Moscow, 1966)», Mir, (1968), 712-719 (See also Bull. A.M.S., 72 (1966), 225-231 and 736). Zbl0141.40602
  2. [2] W. BROWDER, Free Zp-actions on homotopy spheres, In «Topology of manifolds» (Proceedings of the 1969 Georgia Conference on Topology of Manifolds), Markham Press, Chicago (1970). Zbl0284.57030
  3. [3] S. E. CAPPELL, Superspinning and knot complements, In «Topology of manifolds» (Proceedings of the 1969 Georgia Conference on Topology of Manifolds), Markham Press, Chicago (1970). Zbl0281.57001
  4. [4] S. E. CAPPELL, A splitting theorem for manifolds and surgery groups, Bull. A.M.S., 77 (1971), 281-286. Zbl0215.52601MR44 #2234
  5. [5] S. E. CAPPELL, Lecture notes on the splitting theorem, Mimeo, notes Princeton University, 1972. 
  6. [6] S. E. CAPPELL and J. L. SHANESON, Submanifolds, group actions and knots I. Bull. A.M.S., to appear. 
  7. [7] S. E. CAPPELL and J. L. SHANESON, Submanifolds, group actions and knots II, Bull. A.M.S., to appear. Zbl0263.57013
  8. [8] S. E. CAPPELL and J. L. SHANESON, Topological knots and cobordism, Topology, to appear. Zbl0268.57006
  9. [9] S. E. CAPPELL and J. L. SHANESON, The placement problem in codimension two and homology equivalent manifolds, to appear. Zbl0279.57011
  10. [10] S. E. CAPPELL and J. L. SHANESON, Non-Locally flat embeddings, Bull. A.M.S., to appear. Zbl0265.57005
  11. [11] F. T. FARRELL and W. C. HSIANG, Manifolds with π1 = Gα × T, to appear (See also Bull. A.M.S., 74 (1968), 548-553). 
  12. [12] R. FOX and J. MILNOR. Singularities of 2-spheres in 4-space, Bull. A.M.S., 63 (1965), 406. Zbl0146.45501
  13. [13] M. A. KERVAIRE, Les nœuds de dimension supérieure, Bull. Soc. Math. de France, 93 (1965), 225-271. Zbl0141.21201MR32 #6479
  14. [14] J. LEVINE, Knot cobordism in codimension two, Comm. Math. Helv., 44 (1968), 229-244. Zbl0176.22101MR39 #7618
  15. [15] J. LEVINE, Invariants of knot cobordism, Inventiones Math, 8 (1969), 98-110. Zbl0179.52401MR40 #6563
  16. [16] S. LOPEZ DE MEDRANO, «Involutions on Manifolds», Springer-Verlag, (1971). 
  17. [17] S. LOPEZ DE MEDRANO, Invariant knots and surgery in codimension two, Actes du Congrès Int. des Mathématiciens Vol. 2. Gauthier-Villars, Paris, pp. 99-112. Zbl0231.57020MR54 #8653
  18. [18] J. L. SHANESON, Wall's Surgery obstruction groups for Z × G, Ann. of Math., 90 (1969), 296-334. Zbl0182.57303MR39 #7614
  19. [19] J. L. SHANESON, Surgery on 4-manifolds and topological transformation groups, In Procedings of the Amhearst Conference on Transformation groups (1970), to appear. 
  20. [20] C. T. C. WALL, «Surgery on compact manifolds», Academic press, 1970. Zbl0219.57024
  21. [21] L. JONES. Three characteristic classes measuring the obstruction to P.L. local unknottedness, Bull. A.M.S., to appear. Zbl0271.57005

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