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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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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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: http://dx.doi.org/10.1090/S0002-9947-96-01493-6
PII: S 0002-9947(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