Abstract
Two theorems due to V. A. Rokhlin are proved: the theorem on the third stable homotopy group of spheres: \(\pi _{n + 3} (S^n ) \approx \mathbb{Z}_{24} {\text{ }}for{\text{ }}n \geqslant {\text{5}}\); and the theorem on the divisibility by 16 of the signature of a spin 4-manifold. The proofs use immersion theory. Bibliography 17 titles.
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REFERENCES
T. Ekholm, “Differential 3-knots in 5-spaces with and without self intersections," to appear in Topology.
M. Gromov, Partial Differential Relations (Ergeb. Math. Grenzgebiete, 3. Folge, Band 9), Springer-Verlag, Berlin-New York (1986).
L. Guillou and A. Marin (eds.), A la Recherche de la Topologie Perdue (Progress Math., 62), Birkh¨ auser, Boston-Basel-Stuttgart (1986).
R. Gompf and A. Stipsicz, 4-Manifolds and Kirby Calculus (Grad. Stud. Math., 20), Amer. Math. Soc., Providence (1999).
M. Hirsch, “Immersions of manifolds,” Trans. Amer. Math. Soc., 93, 242–276(1959).
J. F. Hughes and P. M. Melvin, “The Smale invariant of a knot,” Comment. Math. Helv., 60, 615–627(1985).
D. Husemoller, Fibre Bundles, McGraw-Hill Book Company, New York (1966).
R. Kirby, “The topology of 4-manifolds,” Lect. Notes Math., 1374(1989).
U. Koschorke, “Multiple points of immersions and the Kahn-Priddy theorem,” Math. Z., 169, 223–252(1979).
S. Levy, Making Waves, A Guide to the Ideas behind Outside In, Geometry Center, University of Minnesota (1989).
Y. Matsumoto, “Elementary proof of Rohlin's theorem on signature and of its generalization due to Guillou and Marin,” in: A la Recherche de la Topologie Perdue, Birkhauser, Boston-Basel-Stuttgart (1986).
J. Milnor, Morse Theory (Ann. Math. Studies, 51), Princeton University Press, Princeton, New Jersey (1963).
U. Pinkall, “Regular homotopy classes of immersed surfaces,” Topology, 24, 421–434(1985).
C. Rourke and B. Sanderson, “A new approach to immersion theory,” in: Proc. 6th G¨okova Geometry-Topology Conf. (1998).
S. Smale, “The classiffication of immersions of spheres in Euclidean spaces,” Ann. Math, 327–344(1959).
A. Stipsicz, “On the vanishing of the third spin cobordism group Spin 3,” in this volume.
R. Wells, “Cobordisms of immersions,” Topology, 5, 281–294(1966).
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Szűcs, A. Two Theorems of Rokhlin. Journal of Mathematical Sciences 113, 888–892 (2003). https://doi.org/10.1023/A:1021208007146
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DOI: https://doi.org/10.1023/A:1021208007146