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

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