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Non-embedding theorems for Y-spaces

Published online by Cambridge University Press:  24 October 2008

K. H. Mayer  and
R. L. E. Schwarzenberger
K. H. Mayer
Affiliation:
University of Bonn and University of Warwick
R. L. E. Schwarzenberger
Affiliation:
University of Bonn and University of Warwick
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Extract

Let X be a compact differentiable manifold of dimension 2m. A differentiable map from X to euclidean (2m + t)-space is an immersion if its Jacobian has rank 2m at each point of X; it is an embedding if it is also one–one. The existence of such an embedding or immersion implies that the characteristic classes of X satisfy certain integrality conditions; these can be used to obtain lower bounds for the integer t. In a similar way many other geometric properties of X can be deduced from a single integrality theorem involving characteristic classes of various vector bundles over X (see for instance (5)).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 3 , July 1967 , pp. 601 - 612
Copyright
Copyright © Cambridge Philosophical Society 1967

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References

REFERENCES

(1)Adams, J. F.Vector fields on spheres. Ann. of Math. 75 (1962), 603–632.Google Scholar
(2)Atiyah, M. F. and Hirzebruch, F.Riemann–Roch theorems for differentiable manifolds. Bull. American Math. Soc. 65 (1958), 276–281.CrossRefGoogle Scholar
(3)Atiyah, M. F. and Hirzebruch, F.Quelques théorèmes de non-plongement pour les variétés différentiables. Bull. Soc. Math. France 87 (1959), 383–396.Google Scholar
(4)Littlewood, D. E.The theory of group characters and matrix representations of groups (Oxford University Press, 1940).Google Scholar
(5)Mayer, K. H.Elliptische Differentialoperatoren und Ganzzahligkeitssätze für charakteristische Zahlen. Topology 4 (1965), 295–313.CrossRefGoogle Scholar
(6)Roberts, R. S.Bundles of Grassmannians and integrality theorems. Proc. Cambridge Philos. Soc. 63 (1967), 45–53.Google Scholar
(7)Sanderson, B. J. and Schwarzenberger, R. L. E.Non-immersion theorems for differentiable manifolds. Proc. Cambridge Philos. Soc. 59 (1963), 319–322.CrossRefGoogle Scholar
(8)Schwarz, W.Zwei Anwendungen topologischer Ganzzahligkeitssätze. Diplomarbeit (Bonn, 1965).Google Scholar
(9)Steer, B.Une interpretation géométrique des nombres de Radon–Hurwitz. Ann. Instit-Fourier Grenoble (to appear).Google Scholar
(10)Weyl, H.The classical groups (Princeton University Press, 1939).Google Scholar