$m\ge 2$ dimensional manifold $M$ to directed surface $F$ general mapping $f: The singular set of M\to F$ is the closed smooth curve $\Sigma(f)$ . Check the parity of the number of components of $\Sigma(f)$. The singular set image $f(\Sigma)$ inherits the normal local orientations via the so-called chessboard function. Such a local orientation gives rise to the cumulative winding number $\omega(f)\in \frac{1}{2}\mathbb{Z}$ of $\Sigma(f)$. We also define an invariant $I(f)$ which is the remainder class modulo $4$ of the sum of the number of components of $\Sigma(f)$ when the manifold $M$ has even dimension. Twice the number of self-intersections of $f(\Sigma)$. Using the cumulative number of turns and the invariant $I(f)$, the parity of the number of connected components of $\Sigma(f)$ is the homotopy of $f$ if any of the following conditions are met: Indicates that it does not change below. (i) the dimension of $M$ is even, (ii) the homotopy singular set is an orientable manifold, or (iii) the image of the homotopy singular set has no triple self-intersection points.
