### 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 R^{2n-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 R^{2n}. We also study generic projections of an embedded nmanifold in R^{2n} into 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 R^{4}. The problem of lifting a map into R^{2n-1} to an embedding into R^{2n} is also studied.

Original language | English |
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Pages (from-to) | 2585-2606 |

Number of pages | 22 |

Journal | Transactions of the American Mathematical Society |

Volume | 348 |

Issue number | 7 |

Publication status | Published - 1996 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

^{2n}and the double points of their generic projections into R

^{2n-1}.

*Transactions of the American Mathematical Society*,

*348*(7), 2585-2606.