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On the cobordism groups of immersions and embeddings. (English) Zbl 0722.57017

In the present paper we suggest as a cobordism invariant of an immersed or embedded submanifold in Euclidean space the singularity set of its projection to a hyperplane. A similar approach has been employed by T. Banchoff [Proc. Am. Math. Soc. 46, 402-406 (1974; Zbl 0309.57016)] and U. Koschorke [Vector fields and vector bundle morphisms - a singularity approach (Lect. Notes Math. 847) (1981; Zbl 0459.57016)]; see also the author’s paper in Lect. Notes Math. 788, 223-244 (1980; Zbl 0442.57017). We consider the range of dimensions \(n\leq 3k\) where n is the dimension and k is the codimension. We prove that in this range (1) our singularity invariant is complete modulo 2-torsion, and (2) modulo- torsion, it can take any value from Thom’s oriented cobordism group of corresponding dimension for k even, while for k odd this invariant is always trivial.

MSC:

57R90 Other types of cobordism
57R40 Embeddings in differential topology
57R42 Immersions in differential topology
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