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Enumeration of immersions of m-manifolds in (2m-2)-manifolds by the singularity method

  • U. Kaiser
  • B. H. Li
Immersions And Vector Bundle Morphisms
Part of the Lecture Notes in Mathematics book series (LNM, volume 1350)

Abstract

We give an enumeration of the set of homotopy classes of monomorphisms αm↪β2m-2 for vectorbundles α,β of dimensions m and (2m-2) over a differentiable closed manifold Mm of dimension m≥6. This is applied to the classification of immersions homotopic to a map g : Mm → N2m−2. In particular we can enumerate immersions from Mm in ℝ2m−2 for all m≥6.

Keywords

Vector Bundle Short Exact Sequence Homotopy Class Obstruction Theory Real Projective Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [ 1]
    BAUSUM, D.R., Embeddings and immersions of manifolds in Euclidean space, Trans. AMS 213 (1975), 263–305MathSciNetCrossRefzbMATHGoogle Scholar
  2. [ 2]
    BECKER, J.C., Cohomology and the classification of liftings, Trans. AMS 133 (1968), 447–475MathSciNetCrossRefzbMATHGoogle Scholar
  3. [ 3]
    DAX, J.P., Étude homotopique des espaces de plongements, Ann. scient. Ec. Norm. Sup. 4e sèrie, t.5 (1972), 303–377MathSciNetzbMATHGoogle Scholar
  4. [ 4]
    GREENBLATT, R., The twisted Bockstein coboundary, Proc. Camb. Phil. Soc. 61 (1965), 295–297MathSciNetCrossRefzbMATHGoogle Scholar
  5. [ 5]
    HIRSCH, M., Immersions of manifolds, Trans. AMS 93 (1959), 242–276.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [ 6]
    JAMES, I., THOMAS, E., Note on the classification of cross-sections, Topology 4 (1966), 351–359MathSciNetCrossRefzbMATHGoogle Scholar
  7. [ 7]
    JAMES, I., THOMAS, E., On the enumeration of cross-sections, Topology 5 (1966), 95–114MathSciNetCrossRefzbMATHGoogle Scholar
  8. [ 8]
    JAMES, I., The topology of Stiefel manifolds, London Math. Soc. Lecture Notes 24 (1976)Google Scholar
  9. [ 9]
    KOSCHORKE, U., Vector fields and other vector bundle morphisms-A singularity approach, Lecture Notes in Math. 847, Springer-Verlag (1981)Google Scholar
  10. [10]
    KOSCHORKE, U., The singularity method and immersions of m-manifolds into manifolds of dimensions 2m − 2, 2m − 3 and 2m − 4, these proceedings volumeGoogle Scholar
  11. [11]
    LARMORE, L.L., RIGDON, R.D., Enumerating immersions and embeddings of projective spaces, Pac. J. of Math. 64 (2) (1976), 471–491MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    LARMORE, L.L., THOMAS, E., Group extensions and twisted cohomology theories, Illinois J. Math. 17 (1973), 397–410MathSciNetzbMATHGoogle Scholar
  13. [13]
    LI, B.H., On immersions of manifolds in manifolds, Scientia Sinica Ser. A (25) (1982), 255–263Google Scholar
  14. [14]
    LI, B.H., On immersions of m-manifolds in (m+1)-manifolds, Math. Z. 182 (1983), 311–320MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    LI, B.H., On classification of immersions of n-manifolds in (2n−1)-manifolds, Comment. Math. Helv. 57 (1982), 135–144MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    LI, B.H., Classification of immersions of m-manifolds in (2m−1) and (2m−2)-manifolds, Northeastern Math. J. 2 (1) (1986), 87–99MathSciNetzbMATHGoogle Scholar
  17. [17]
    LI, B.H., Normal vectorfields of immersions of n-manifolds in (2n-1)-manifolds, Scientia Sinica Ser. A (31) (1988), 31–45Google Scholar
  18. [18]
    LI, B.H., Existence of immersions of n-manifolds in manifolds of dimension 2n−2 or 2n−3, J. Sys. Sci. & Math. Scis. 6 (3) (1986), 177–185zbMATHGoogle Scholar
  19. [19]
    LI, B.H., HABEGGER, N., A two stage procedure for the classification of vectorbundle monomorphisms with applications to the classification of immersions homotopic to a map, Lecture Notes in Math. 1051, Springer-Verlag 1984, 293–314Google Scholar
  20. [20]
    LI, B.H., PETERSON, F.P., On immersions of k-manifolds in (2k−1)-manifolds, Proc. of the AMS 83 (1) (1981), 159–162MathSciNetzbMATHGoogle Scholar
  21. [21]
    LI, B.H., PETERSON, F.P., Immersions of n-manifolds into (2n−2)-manifolds, Proc. of the AMS 97 (3), (1986), 531–538MathSciNetzbMATHGoogle Scholar
  22. [22]
    MAHOWALD, M., On obstruction theory in orientable fiber bundles, Trans. AMS 110 (1964), 315–349MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    MASSEY, W.S., PETERSON, F.P., On the dual Stiefel-Whitney classes of a manifold, Bol. Soc. Mat. Mexicana 8 (1963), 1–13MathSciNetzbMATHGoogle Scholar
  24. [24]
    McCLENDON, J.F., Obstruction theory in fiber spaces, Math. Z. 120 (1971), 1–12MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    OLK, C., Immersionen von Mannigfaltigkeiten in Euklidische Räume, Dissertation Siegen 1980Google Scholar
  26. [26]
    PAECHTER, G.F., The groups πrVn,m(I), Quart. J. of Math. Oxford (2) 7 (1956), 249–268MathSciNetCrossRefGoogle Scholar
  27. [27]
    SAMLSON, H., A note on the Bockstein operator, Proc. AMS 15 (1964), 450–453MathSciNetCrossRefGoogle Scholar
  28. [28]
    SPANIER, E.H., Algebraic topology, McGraw Hill (1966)Google Scholar
  29. [29]
    STEENROD, N., Topology of fiber bundles, Princeton University Press N.J. (1951)CrossRefzbMATHGoogle Scholar
  30. [30]
    WU, W., Les i-carrés dans une variété grassmannienne, C.R. Acad. Sci. Paris 230 (1950), 918–920MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • U. Kaiser
  • B. H. Li

There are no affiliations available

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