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

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 $\text {\bf R}^{2n-1}$, 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.