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Immersed $n$-manifolds in $\text {\bf R}^{2n}$ and the double points
of their generic projections into $\text {\bf R}^{2n-1}$


Authors: Osamu Saeki and Kazuhiro Sakuma
Journal: Trans. Amer. Math. Soc. 348 (1996), 2585-2606
MSC (1991): Primary 57R42; Secondary 57R45, 57R40
DOI: https://doi.org/10.1090/S0002-9947-96-01493-6
MathSciNet review: 1322957
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Abstract: We give two congruence formulas concerning the number of non-trivial double point circles and arcs of a smooth map with generic singularities --- the Whitney umbrellas --- of an $n$-manifold into $\text {\bf R}^{2n-1}$, which generalize the formulas by Szücs for an immersion with normal crossings. Then they are applied to give a new geometric proof of the congruence formula due to Mahowald and Lannes concerning the normal Euler number of an immersed $n$-manifold in $\text {\bf R}^{2n}$. We also study generic projections of an embedded $n$-manifold in $\text {\bf R}^{2n}$ into $\text {\bf R}^{2n-1}$ and prove an elimination theorem of Whitney umbrella points of opposite signs, which is a direct generalization of a recent result of Carter and Saito concerning embedded surfaces in $\text {\bf R}^{4}$. The problem of lifting a map into $\text {\bf R}^{2n-1}$ to an embedding into $\text {\bf R}^{2n}$ is also studied.


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  • 1. M. Audin, Fibrés normaux d'immersions en dimension double, points doubles d'immersions lagrangiennes et plongements totalement réels, Comment. Math. Helv. 63 (1988), 593--623. MR 89m:57032
  • 2. M. Audin, Quelques remarques sur les surfaces lagrangiennes, J. Geom. Phys. 7 (1990), 583--598. MR 92i:57022
  • 3. T. F. Banchoff, Double tangency theorems for pairs of submanifolds, Geometry Symposium Utrecht 1980 (Looijenga, Siersma and Takens, eds.), Lect. Notes in Math., vol. 894, Springer-Verlag, Berlin and New York, 1981, pp. 26--48. MR 83h:53005
  • 4. S. J. Blank and C. Curley, Desingularizing maps of corank one, Proc. Amer. Math. Soc. 80 (1980), 483--486. MR 82e:57017
  • 5. J. S. Carter and M. Saito, Canceling branch points on projections of surfaces in 4-space, Proc. Amer. Math. Soc. 116 (1992), 229--237. MR 93i:57029
  • 6. C. A. Giller, Towards a classical knot theory for surfaces in $\text {\bf R}^{4}$, Illinois J. Math. 26 (1982), 591--631. MR 84c:57011
  • 7. M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Grad. Texts in Math. no.14, Springer-Verlag, New York-Heidelberg-Berlin, 1973. MR 49:6269
  • 8. M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242--276. MR 22:9980
  • 9. S. Kamada, Non-orientable surfaces in 4-space, Osaka J. Math. 26 (1989), 367--385. MR 91g:57022
  • 10. J. Lannes, Sur les immersions de Boy, Algebraic Topology, Aarhus 1982, edited by Madsen and Oliver, Lecture Notes in Math. 1051, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1984, pp. 263--270. MR 86d:57015
  • 11. Li Bang-He, Generalization of the Whitney-Mahowald Theorem, Trans. Amer. Math. Soc. 346 (1994), 511--521.
  • 12. M. Mahowald, On the normal bundle of a manifold, Pacific J. Math. 14 (1964), 1335-- 1341. MR 31:757
  • 13. W. S. Massey, On the Stiefel-Whitney classes of a manifold, Amer. J. Math. 82 (1960), 92--102. MR 22:1918
  • 14. W. S. Massey, Proof of a conjecture of Whitney, Pacific J. Math. 31 (1969), 143--156. MR 40:3570
  • 15. J. Mather, Generic projections, Ann. of Math. 98 (1973), 226--245. MR 50:14835
  • 16. F. Ronga, `La class duale aux points double' d'une application, Compositio Math. 27 (1973), 223--232. MR 52:11936
  • 17. O. Saeki, Notes on the topology of folds, J. Math. Soc. Japan 44 (1992), 551--566. MR 93f:57037
  • 18. A. Szücs, Surfaces in $\text {\bf R}^{3}$, Bull. London Math. Soc. 18 (1986), 60--66. MR 88a:57068
  • 19. A. Szücs, Note on double points of immersions, Manuscripta Math. 76 (1992), 251--256. MR 93h:57047
  • 20. R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17--86. MR 15:890a
  • 21. H. Whitney, On the topology of differentiable manifolds, Lectures in Topology, Michigan Univ. Press, 1940. MR 3:133a
  • 22. H. Whitney, The general type of singularity of a set of $2n - 1$ smooth functions of $n$ variables, Duke Math. J. 10 (1943), 161--172. MR 4:193b
  • 23. H. Whitney, The self-intersection of a smooth $n$-manifold in $2n$-space, Ann. of Math. 45 (1944), 220--246. MR 5:273g
  • 24. H. Whitney, The singularities of a smooth $n$-manifold in $(2n-1)$-space, Ann. of Math. 45 (1944), 247--293. MR 5:274a
  • 25. Y. Yamada, An extension of Whitney's congruence, Osaka J. Math. 32 (1995), 185--192.

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Additional Information

Osamu Saeki
Affiliation: Department of Mathematics, Faculty of Science, Hiroshima University, Higashi-Hiroshima 739, Japan
Email: saeki@top2.math.sci.hiroshima-u.ac.jp

Kazuhiro Sakuma
Affiliation: Department of General Education, Kochi National College of Technology, Nankoku City, Kochi 783, Japan
Email: sakuma@cc.kochi-ct.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-96-01493-6
Keywords: Double point circle, Whitney umbrella, normal Euler number, generic projection
Received by editor(s): November 29, 1994
Additional Notes: The first author was partially supported by CNPq, Brazil.
Article copyright: © Copyright 1996 American Mathematical Society

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