Cross-cap singularities counted with sign

Abstract

A method for computing the algebraic number

of cross-cap singularities for mapping from m-dimensional com- pact manifold with boundary M ⊂ R

m into R 2m−1 , m is odd, is

presented. As an application, the intersection number of an immer- sion g : S

m−1 (r) → R 2m−2 is described as the algebraic number of

cross-caps of a mapping naturally associated with g.