Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Note on the Codimension Two Splitting Problem

  • Received : 2016.06.23
  • Accepted : 2016.11.11
  • Published : 2019.09.23
Download PDF
⟨ Previous Next ⟩

Abstract

Let W and V be manifolds of dimension m + 2, M a locally flat submanifold of V whose dimension is m. Let $f:W{\rightarrow}V$ be a homotopy equivalence. The problem we study in this paper is the following: When is f homotopic to another homotopy equivalence $g:W{\rightarrow}V$ such that g is transverse regular along M and such that $g{\mid}g^{-1}(M):g^{-1}(M){\rightarrow}M$ is a simple homotopy equivalence? $L{\acute{o}}pez$ de Medrano (1970) called this problem the weak h-regularity problem. We solve this problem applying the codimension two surgery theory developed by the author (1973). We will work in higher dimensions, assuming that $$m{\geq_-}5$$.

Keywords

References

  1. H. Bass (ed.), Algebraic K-theory, III: Hermitian K-theory and geometric applications, Lecture Noes in Math., 343, Springer-Verlag, 1973.
  2. V. Blanloeil, Y. Matsumoto, and O. Saeki, Pull back relation for non-spherical knots, J. Knot Theory Ramifications, 13(2004), 689-701. https://doi.org/10.1142/S0218216504003378
  3. S. E. Cappell, Groups of singular Hermitian forms, Algebraic K-theory, III: Hermitian K-theory and geometric applications, 513-525, Lecture Noes in Math., 343, Springer-Verlag, 1973.
  4. S. E. Cappell and J. L. Shaneson, Submanifolds, group actions and knots. I, II, Bull. Amer. Math. Soc. 78(1972), 1045-1048. https://doi.org/10.1090/S0002-9904-1972-13103-3
  5. S. E. Cappell and J. L. Shaneson, Submanifolds, group actions and knots. I, II, Bull.Amer. Math. Soc.. 78(1972), 1049-1052. https://doi.org/10.1090/S0002-9904-1972-13106-9
  6. S. E. Cappell and J. L. Shaneson, The codimension two placement problem and homology equivalent manifolds, Ann. of Math., 99(1974), 277-348. https://doi.org/10.2307/1970901
  7. S. E. Cappell and J. L. Shaneson, Fundamental groups, $\Gamma$-groups, and codimension two submanifolds, Comment. Math. Helv., 51(1976), 437-446. https://doi.org/10.1007/BF02568168
  8. M. H. Freedman, Surgery on codimension 2 submanifolds, Mem. Amer. Math. Soc., 12(191)(1977), 93 pp.
  9. M. Kato and Y. Matsumoto, Simply connected surgery of submanifolds in codimension two, I, J. Math. Soc. Japan, 24(1972), 586-608. https://doi.org/10.2969/jmsj/02440586
  10. M. Kato (ed.), Some problems in Topology, In: Manifolds Tokyo 1973 (ed. Akio Hattori), University of Tokyo Press, 1975.
  11. H. J. Kim and D. Ruberman, Topological spines of 4-manifolds, https://arxiv.org/abs/1905.03608, 2019.
  12. J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv., 44(1969), 229-244. https://doi.org/10.1007/BF02564525
  13. J. Levine, Invariants of knot cobordism, Invent. Math., 8(1969), 98-110. https://doi.org/10.1007/BF01404613
  14. A. S. Levine and T. Lidman, Simply-connected, spineless 4-manifolds. https://arxiv.org/abs/1803.01765, 2018.
  15. S. Lopez de Medrano, Invariant knots and surgery in codimension 2, Proc. ICM Nice, 2(1970), 99-112.
  16. Y. Matsumoto, Hauptvermutung for ${\pi}_1=\mathbb{Z}$, J. Fac. Sci. Univ. Tokyo, Sec. IA, 16(1969), 165-177.
  17. Y. Matsumoto, Note on the weak h-regularity problem, Unpublished Note, The University of Tokyo, (circa 1973).
  18. Y. Matsumoto, Knot cobordism groups and surgery in codimension two, J. Fac. Sci. Univ. Tokyo, Sec. IA, 20(1973), 253-317.
  19. Y. Matsumoto, Some relative notions in the theory of Hermitian forms, Proc. Japan Acad., 49(1973), 583-587. https://doi.org/10.3792/pja/1195519220
  20. Y. Matsumoto, A 4-manifold which admits no spine, Bull. Amer. Math. Soc., 81(1975), 467-470. https://doi.org/10.1090/S0002-9904-1975-13787-6
  21. Y. Matsumoto, Some counterexamples in the theory of embeddig manifolds in codimension two, Sci. Papers College Gen. Ed. Univ. Tokyo, 25(1975), 49-57.
  22. Y. Matsumoto, On the equivalence of algebraic formulations of knot cobordism, Japan. J. Math., 3(1977), 81-103. https://doi.org/10.4099/math1924.3.81
  23. Y. Matsumoto, Wild embeddings of piecewise linear manifolds in codimension two, Geometric Topology, 393-428, Academic Press, New York, 1979.
  24. J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc., 72(1966), 358-426. https://doi.org/10.1090/S0002-9904-1966-11484-2
  25. J. Milnor, On isometries of inner product spaces, Invent. Math., 8(1969), 83-97. https://doi.org/10.1007/BF01404612
  26. J. Milnor and D. Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete 73, Springer-Verlag, 1973.
  27. I. Namioka, Maps of pairs in homotopy theory, Proc. London Math. Soc. (3), 12(1962), 725-738. https://doi.org/10.1112/plms/s3-12.1.725
  28. A. Ranicki, Algebraic L-theory, I: Foundations, Proc. London Math. Soc. (3), 27(1973), 101-125. https://doi.org/10.1112/plms/s3-27.1.101
  29. A. Ranicki, Exact sequences in the algebraic theory of surgery, Math. Notes 26, Princeton University Press, Princeton, N.J., 1981.
  30. A. Ranicki, High-dimensional knot theory: Algebraic surgery in codimension 2, Springer Monographs in Math., Springer-Verlag, New York, 1998.
  31. J. L. Shaneson, Wall's surgery obstruction groups for $G{\times}{\mathbb{Z}}$, Ann. of Math., 90(1969), 296-334. https://doi.org/10.2307/1970726
  32. J. L. Shaneson, Hermitian K-theory in topology, Algebraic K-theory, III: Hermitian K-theory and geometric applications, 1-40, Lecture Noes in Math., 343, Springer-Verlag, 1973.
  33. C. T. C. Wall, Surgery on compact manifolds, London Mathematical Society Mono-graphs 1, Academic Press, New York, 1970.