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 \(M\subset \mathbb{R}^m\) into \(\mathbb{R}^{2m-1}\), \(m\) is odd, is presented. As an application, the intersection number of an immersion \(g\colon S^{m-1}(r)\to\mathbb{R}^{2m-2}\) is described as the algebraic number of cross-caps of a mapping naturally associated with \(g\).