Immersions and their self intersections

Sammanfattning: Regular homotopy classes of immersions of the k-dimensional sphere Sk into (k + n)-dimensional Euclidean space Rk+n are known to form finitely generated Abelian groups. It is known how to calculate the regular homotopy class of a generic immersion Sk → R2k in terms of its self intersection. A formula for the regular homotopy class of a generic immersion Sk → R2k-r, r = 1, 2 and k ≥ 4, is found. It involves only terms depending on the geometry of the self intersection of the generic immersion. It is shown that, for immersions S3 → R5, such a formula does not exist. More precisely, regular homotopy classes of immersions S3 → R5 constitute an infinite cyclic group. The classes containing embeddings form a subgroup. It is proved that this subgroup has index 24 and the obstruction for a generic immersion to be regularly homotopic to an embedding is expressed in terms of geometric invariants of its self intersection. It is shown that, up to regular homotopy through generic immersions, a generic immersion in the metastable range of a sufficiently high-connected manifold into Euclidean space depends only on the geometry of its self intersection. In the cases when the self intersection has dimension 0, 1, or 2, numerical invariants of self intersections are found and provedto give a complete classification of generic immersions. All finite order invariants of generic immersions Sk → R2k-r, r = 1, 2 and k ≥ 2r + 2 are found. It is showed that they are not sufficient to separate immersions which cannot be deformed into each other by regular homotopy through generic immersions. Two independent first order invariants J and St of generic immersions S3; → R5; are constructed. It is proved that any first order invariant is a linear combination of these.