Geodesic
Self-intersection surfaces, regular homotopy, and finite order invariants
Algebra i analiz, Tome 11 (1999) no. 5, pp. 250-272.

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Explicit formulas for the regular homotopy classes of generic immersions $S^k\to{\mathbb R}^{2k-2}$ are given in terms of the corresponding self-intersection manifolds with natural additional structures. There is a natural notion of finite order invariants of generic immersions. We determine the group of $m$th order invariants for each $m$ and prove that the finite order invariants are not sufficient for distinguishing generic immersions that cannot be obtained from each other by a regular homotopy through generic immersions.
Keywords: immersion, regular homotopy, finite order invariants, spin and pin structures. 
@article{AA_1999_11_5_a11,
     author = {T. Ekholm},
     title = {Self-intersection surfaces, regular homotopy, and finite order invariants},
     journal = {Algebra i analiz},
     pages = {250--272},
     publisher = {mathdoc},
     volume = {11},
     number = {5},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AA_1999_11_5_a11/}
}
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T. Ekholm. Self-intersection surfaces, regular homotopy, and finite order invariants. Algebra i analiz, Tome 11 (1999) no. 5, pp. 250-272. http://geodesic.mathdoc.fr/item/AA_1999_11_5_a11/