Disentanglements of maps from \(2n\)-space to \(3n\)-space
For a finitely \({\mathcal A}\)-determined map-germ \(f:(\mathbb C^{3n},S)\to (\mathbb C^{2n},0)\), where \(S\) is a finite set, the author studies the first and second disentanglement of \(f\) [see for instance \textit{W. L. Marar}, Manuscr. Math. 80, No.~3, 273--281 (1993; Zbl 0793.32018); \textit{V. Goryunov} and \textit{D. Mond}, Compos. Math. 89, No.~1, 45--80 (1993; Zbl 0839.32017)]. He proves that the first disentanglement of \(f\) is homotopically equivalent to a wedge of \(n+1\)-spheres and \(2\)-spheres. This corresponds to what happens in general for monogerms [\textit{K. Houston}, Topology 36, No.~5, 1077--1121 (1997; Zbl 0877.58010)]. One of the main tools that the author uses is an image computing spectral sequence. For corank one monogerms the second disentanglement is shown to be homotopically equivalent to a wedge of \(n\)-spheres and circles. The author also generalizes the singularities given by \(f(x,y)=(x,y^3,xy+y^{3k-1})\), in which each member in its \(H_k\) series has a good real perturbation, to \(2n\)-space to \(3n\)-space situation and shows that not only do the maps have good real perturbations but the real and complex \(r\)-th disentanglements for \(r=1,2,3\) are in fact homotopically equivalent.
scientific article; zbMATH DE number 5117284