The homotopy principle for maps with singularities of given \(\mathcal K\)-invariant class (Q2370113)

This interesting paper is organized into seven sections. In Section 1 the author explains the notations which are used in the paper. In Section 2 the definitions and the fundamental properties of K-orbits are reviewed. In Theorem 3.2 of Section 3, a special form of a homotopy principle is stated. In Section 4 the author gives two lemmas which are used to deform the section \(s\) in a nice position and in Section 5 he constructs an O-regular map around the singularities of a given type. The main result of the paper, contained in Theorem 3.2, is proved in Section 6. In the last section some applications to maps with K-k-simple singularities of given class are presented. These results are connected to other author's papers such as [J. Math. Soc. Japan 37, 471--487 (1985; Zbl 0581.58005); Hokkaido Math. J. 20, No. 3, 571--578 (1991; Zbl 0751.57012); J. Math. Kyoto Univ. 41, 691--735 (2002); Jap. J. Math. 30, 29--73 (2004; Zbl 1070.57018); Trans. Am. Math. Soc. 359, No.~2, 489--515 (2007; Zbl 1122.58023)].