The author gives three connecting lemmas which generalize results of V. A. Rokhlin [Math. Dokl. 11, 316-319 (1970); translation from Dokl. Akad. Nauk SSSR 191, 27-29 (1970; Zbl 0214.22603)] and are used in a manner similar to that of the reviewer [Mich. Math. J. 34, 85-91 (1987; Zbl 0624.57019)] to reduce questions about the minimal genus of an embedded oriented surface and the normal Euler number of an embedded non-orientable surface to questions about embedded spheres or \((D^2,S^1)\). This leads to new results on these questions, as well as applications to knot theory. Some results depend on the 11/8 conjecture. Using Furuta’s proof of a slightly weaker form on this [Monopole equation and the 11/8 conjecture, preprint], one can modify these statements to give new results independent of this conjecture. Recent papers by Ruberman [Proc. Gökova Geom. Top. Conf. 1995, 129-133] and Li and Li [Minimal genus embeddings in \(S^2\times S^2\) and \(CP^2 \# n \overline{CP^2}\) with \(n \leq 8\) (preprint)] has improved results given here for the minimal genus in a rational surface.