Cross-cap singularities counted with sign

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 $g$.