On bordism and cobordism groups of Morin maps

Abstract

The abstract link L d of the complex isolated singularity x 2 + y 2 + z 2 + v 2d = 0 in (C 4 , 0) is diffeomorphic to S 3 × S 2 . We classify the embedded links of these singularities up to regular homotopies precomposed with diffeomorphisms of S 3 × S 2 . Let us denote by i d the inclusion L d ⊂ S 7 . We show that for arbitrary diffeomorphisms ϕ d : S 3 × S 2 −→ L d the compositions i d • ϕ d are image regularly homotopic for two values d 1 and d 2 of d if and only if d 1 ≡ d 2 mod 2.

On certain complex surface singularities

“…A similar question is studied in [21] about the link of (X k , 0) = f −1 k (0) ⊂ (C 4 , 0), where f k (x, y, z, w) = x 2 + y 2 + z 2 + w k . Its link depends on the parity of k, that is…”

“…We mention here two results about a very similar topic to the material discussed in Chapter 3. The articles [9] and [21] study links of isolated hypersurface singularities (cf. Chapter 4) up to regular homotopy.…”

“…A well-defined notion is the image regular homotopy type: the immersions f and g S 3 × S 2 S 7 are image regular homotopic if there is a self diffeomorphism φ of S 3 × S 2 such that g is regular homotopic with f • φ. By [21] there are two image regular homotopy classes of immersions S 3 × S 2 S 7 , and the image regular homotopy class of the inclusion K 2d ⊂ S 7 depends on the parity of d.…”