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Orientable surfaces in the 4-sphere associated with non-orientable knotted surfaces

Published online by Cambridge University Press:  24 October 2008

Seiichi Kamada
Seiichi Kamada
Affiliation:
Department of Mathematics, Osaka City University, Osaka, 558, Japan
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Extract

Let F be a closed connected and non-orientable surface smoothly embedded in the 4-sphere S4 with normal Euler number e(F) = 0. We note that if e(F) = 0, then the non-orientable genus n is even (ef. [7]) and the tubular neighbourhood N(F) of F in S4 which is a D2-bundle over F has a trivial I-subbundle. Let τ be a trivial I-subbundle of N(F) and let τ* = F × I ⊂ N(F) be its orthogonal I-subbundle which is twisted. Then is a closed connected genus n – 1 orientable surface smoothly embedded in S4 and doubly covers F. We call this surface a doubled surface of F in S4 (associated with τ). If a trivial I-subbundle τ is given, then we see that the knot type of F* ⊂ S4 is uniquely determined.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 2 , September 1990 , pp. 299 - 306
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Asano, K.. A note on surfaces in 4-spheres. Math. Sem. Notes Kobe Univ. 4 (1976), 195–198.Google Scholar
[2]Hosokawa, F. and Kawauchi, A.. Proposal for unknotted surfaces in four-space. Osaka J. Math. 16 (1979), 233–248.Google Scholar
[3]Kamada, S.. On doubled surfaces of nonorientablė surfaces in the 4-sphere. Kobe J. Math. (To appear.)Google Scholar
[4]Kawauchi, A.. Three dualities on the integral homology of infinite cyclic coverings of manifolds. Osaka J. Math. 23 (1986), 633–651.Google Scholar
[5]Levine, J.. Polynomial invariants of knots of codimension two. Ann. of Math. 84 (1966), 537–554.CrossRefGoogle Scholar
[6]Litherland, R. A.. The second homology of the group of a knotted surface. Quart. J. Math. Oxford Ser. (2) 32 (1981), 425–434.CrossRefGoogle Scholar
[7]Massey, W. S.. Proof of a conjecture of Whitney. Pacific J. Math. 31 (1969), 143–156.CrossRefGoogle Scholar