Immersed n-manifolds in R2n and the double points of their generic projections into R2n-1

Osamu Saeki, Kazuhiro Sakuma

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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 R2n-1, which generalize the formulas by Sziics 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 R2n. We also study generic projections of an embedded nmanifold in R2n into R2n-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 R4. The problem of lifting a map into R2n-1 to an embedding into R2n is also studied.

Original languageEnglish
Pages (from-to)2585-2606
Number of pages22
JournalTransactions of the American Mathematical Society
Volume348
Issue number7
Publication statusPublished - 1996
Externally publishedYes

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All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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