Immersions without triple points
by
Konstantin Salikhov
Moscow State University

Here we generalize the fundamental Haefliger's theorem [Hae 63] on embeddings to the case of simple immersions (i.e. immersions without triple points). In fact, under the restriction 4(n+1) <= 3m on the dimensions of DIFF manifolds without boundary Vⁿ and M^m (Vⁿ is assumed to be compact), we reduce the problem to Vⁿ be simply immersible into M^m to the purely homotopy-theoretic question. One of the corroloraries from our results is that under the restriction 3n-2m <= 1 (below the metastable range !) the existence of TOP simple immersion Vⁿ\hookrightarrow R^m implies the existence of DIFF simple immersion Vⁿ\hookrightarrow R^m. Note that the possibility of a theorem on simple immersion of such type was claimed in [GE 71], [Gr 86] and [Sz 82]. Our technique goes back to the Haefliger's and Hirsch's original proofs [Hae 63], [HH 62].

[GE 71] M. Gromov and J. Eliashberg, Removal singularities of smooth mappings, Izv. Akad. Nauk SSSR 35:3 (1971), 600-629

[Gr 86] M. Gromov, Partial Differential Relations, Springer-Verlag, Berlin - New York, 1986

[Hae 63] A. Haefliger, Plongements différentiables dans le domaine stable, Comment. Math. Helv. 37 (1963), 155-176

[HH 62] A. Haefliger and M. Hirsch, Immersions in the stable range, Annals of Math. 75 (1962), 231-241

[Sz 82] A. Szücs, The Gromov-Eliashberg proof of the Haefliger's theorem, Studia Sci. Math. Hungarica 17 (1982), 303-318

Date received: June 15, 1999