Cross-cap singularities counted with sign

Abstract

A method for computing the algebraic number of cross-cap singularities for mapping from m-dimensional compact manifold with boundary M ⊂ Rᵐ into R²ᵐ⁻¹, m is odd, is presented. As an application, the intersection number of an immersion g : Sᵐ⁻¹ (r) → R²ᵐ⁻² is described as the algebraic number of cross-caps of a mapping naturally associated with g.