Surfaces Embedded in M 2 × S 1

Jaco, William

Canadian journal of mathematics, Tome 22 (1970) no. 3, pp. 553-568

Résumé

In this paper we study incompressible and injective (see § 2 for definitions) surfaces embedded in M 2 × S 1, where M 2 is a surface and S 1 is the 1-sphere. We are able to characterize embeddings which are incompressible in M 2 × S 1 when M 2 is closed and orientable. Namely, a necessary and sufficient condition for the closed surface F to be incompressible in M 2 × S 1, where M 2is closed and orientable, is that there exists an ambient isotopy ht , 0 ≦ t ≦ 1, of M 2 × S 1onto itself so that either(i) there is a non-trivial simple closed curve J ⊂ M 2 and h 1(F) = J × S 1, or(ii) p\h 1 (F) is a covering projection of h 1 (F) onto M 2, where p is the natural projection of M 2 × S 1onto M 2.

Détail

DOI : 10.4153/CJM-1970-063-x

Bibliographie

[1] 1. Bredon, G. E. and Wood, J. W., Non-orientable surfaces in orientable 3-manifolds, Invent. Math. 7 (1969), 83–110. Google Scholar

[2] 2. Burde, G. and Zieschang, H., A topological classification of certain Z-manifolds, Bull. Amer. Math. Soc. 74 (1968), 122–124. Google Scholar

[3] 3. Greendlinger, M., A class of groups all of whose elements have trivial centralizers, Math. Z. 78 (1962), 91–96. Google Scholar

[4] 4. Hilton, P. J. and Wylie, S., Homology theory: An introduction to algebraic topology (Cambridge Univ. Press, New York, 1960). Google Scholar

[5] 5. Hurewicz, W. and Wallman, H., Dimension theory, rev. éd., Princeton Mathematical Series, Vol. 4 (Princeton Univ. Press, Princeton N.J., 1948). Google Scholar

[6] 6. William, Jaco, On certain subgroups of the fundamental group of a closed surface, Proc. Cambridge Philos. Soc. 67 (1970), 17–18. Google Scholar

[7] 7. William, Jaco and McMillan, D. R. Jr., Retracting three-manifolds onto finite graphs, Illinois J. Math. 14 (1970), 150–158. Google Scholar

[8] 8. Kurosh, A. G., The theory of groups, 2nd éd., Vols. I and II (Chelsea, New York, 1960). Google Scholar

[9] 9. William, Massey, Algebraic topology: An introduction (Harcourt, Brace & World, Inc., New York, 1967). Google Scholar

[10] 10. Lee, Neuwirth, The algebraic determination of the genus of knots, Amer. J. Math. 82 (1960), 791–798. Google Scholar

[11] 11. Lee, Neuwirth, A topological classification of certain 3-manifolds, Bull. Amer. Math. Soc. 69 (1963), 372–375. Google Scholar

[12] 12. Peter, Orlik On certain fiber ed S-manifolds (to appear). Google Scholar

[13] 13. Stallings, J., On the loop theorem, Ann. of Math. (2) 72 (1960), 12–19. Google Scholar

[14] 14. Stallings, J., On fibering certain S-manifolds, pp. 95-100 in Topology of S-manifolds and related topics (Proc. the Univ. of Georgia Institute, 1961), edited by Fort, M. K. (Prentice-Hall, Englewood Cliffs, N.J., 1962). Google Scholar

[15] 15. Waldhausen, F., Gruppen mit Zentrum und Z-dimensionale Mannigfaltigkeiten, Topology 6 (1967), 505–517. Google Scholar

[16] 16. Waldhausen, F., On irreducible S-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. Google Scholar

[17] 17. Zeeman, E. C., Seminar on combinatorial topology, (mimeographed notes) (Publ. Inst. Hautes Etudes Sci., Paris, 1963). Google Scholar