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On Embedding Essential Annuli in M3

Published online by Cambridge University Press:  20 November 2018

C. D. Feustel
C. D. Feustel*
Affiliation:
Virginia Polytechnic Institute and State University, Blacksburg, Virginia
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In [7; 8; 11] it is shown that an appropriate map of a planar surface into a 3-manifold can be replaced by an embedding. In [1 ; 4 ; 6 ; 7 ; 9 ; 10] conditions are given so that a "non-trivial" map of a planar surface (2-sphere) can be replaced by a non-trivial embedding of a planar surface (2-sphere). In this paper we give conditions on an annular map which guarantee the existence of a non-trivial embedding of an annulus. It is reported that F. Waldhausen has proved a similar but stronger "annulus theorem".

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 6 , December 1974 , pp. 1341 - 1350
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Epstein, D. B. A., Projective planes in 3-manifolds, Proc. London Math. Soc. 11 (1961), 469–484.Google Scholar
2. Daigle, R. J. and Feustel, C. D., Essential maps and embeddings in non-orientable M3 , Proc. Amer. Math. Soc. (to appear).Google Scholar
3. Feustel, C. D., Homotopic arcs are isotopic, Proc. Amer. Math. Soc. 17 (1966), 891–896.Google Scholar
4. Feustel, C. D., On embedding annuli in M3, Proc. Amer. Math. Soc. 35 (1972), 581–583.Google Scholar
5. Haken, W., Some results on surfaces in 3-manifolds, Studies in Modern Topology, Math. Assoc. Amer, (distributed by Prentice Hall, Englewood Cliffs, N.J., 1968), 39–98.Google Scholar
6. Papakyriakopoulos, C. D., On solid tori, Proc. London Math. Soc. 7 (1957), 281–299.Google Scholar
7. Papakyriakopoulos, C. D., On Dehn's lemma and the asphericity of knots, Ann. of Math. 66 (1957), 1–26.Google Scholar
8. Shapiro, A. and J. H. C. Whitehead, A proof and extension of Dehn's lemma, Bull. Amer. Math. Soc. 64 (1958), 174–178.Google Scholar
9. Stallings, J., On the loop theorem, Ann. of Math. 72 (1960), 12–19.Google Scholar
10. Whitehead, J. H. C., On 2-spheres in 3-manifolds, Bull. Amer. Math. Soc. 64 (1958), 161–166.Google Scholar
11. Waldhausen, F., Eine Verallgemeinerung des Schleifensatzes, Topology 6 (1967), 501–504.Google Scholar