Immersed n-manifolds in R²ⁿ and the double points of their generic projections into R^2n-1

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 R²ⁿ. We also study generic projections of an embedded nmanifold in R²ⁿ 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 R⁴. The problem of lifting a map into R^2n-1 to an embedding into R²ⁿ is also studied.