Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Nonorientable surfaces in some non-Haken $ 3$-manifolds


Author: J. H. Rubinstein
Journal: Trans. Amer. Math. Soc. 270 (1982), 503-524
MSC: Primary 57N10; Secondary 57N37, 57R95
DOI: https://doi.org/10.1090/S0002-9947-1982-0645327-7
MathSciNet review: 645327
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If a closed, irreducible, orientable $ 3$-manifold $ M$ does not possess any $ 2$-sided incompressible surfaces, then it can be very useful to investigate embedded one-sided surfaces in $ M$ of minimal genus. In this paper such $ 3$-manifolds $ M$ are studied which admit embeddings of the nonorientable surface $ K$ of genus $ 3$. We prove that a $ 3$-manifold $ M$ of the above type has at most $ 3$ different isotopy classes of embeddings of $ K$ representing a fixed element of $ {H_2}(M,\,{Z_2})$. If $ M$ is either a binary octahedral space, an appropriate lens space or Seifert manifold, or if $ M$ has a particular type of fibered knot, then it is shown that the embedding of $ K$ in $ M$ realizing a specific homology class is unique up to isotopy.


References [Enhancements On Off] (What's this?)

  • [1] Joan S. Birman and D. Chillingworth, On the homeotopy group of a non-orientable surface, Proc. Cambridge Philos. Soc. 71 (1972), 437-448. MR 45 #9334. MR 0300288 (45:9334)
  • [2] Joan S. Birman and J. H. Rubinstein, Homeotopy groups of some non-Haken $ 3$-manifolds (to appear).
  • [3] Joan S. Birman, F. Gonzales-Acuna and J. M. Montesinos, Heegaard splittings of prime $ 3$-manifolds are not unique, Michigan Math. J. 23 (1976), 97-103. MR 55 #4177. MR 0431175 (55:4177)
  • [4] G. Bredon and J. Wood, Non-orientable surfaces in orientable $ 3$-manifolds, Invent. Math. 7 (1969), 83-110. MR 39 #7616. MR 0246312 (39:7616)
  • [5] M. Culler, W. Jaco and J. H. Rubinstein, J. London Math. Soc. (to appear).
  • [6] J. Milnor, Lectures on the $ h$-cobordism theorem, Math. Notes, Princeton Univ. Press, Princeton, N.J., 1965. MR 32 #8352. MR 0190942 (32:8352)
  • [7] P. Orlik, Seifert manifolds, Lecture Notes in Math., vol. 291, Springer-Verlag, Berlin and New York, 1972. MR 54 #13950. MR 0426001 (54:13950)
  • [8] J. H. Rubinstein, One-sided Heegaard splittings of $ 3$-manifolds, Pacific J. Math. 76 (1978), 185-200. MR 0488064 (58:7635)
  • [9] -, On $ 3$-manifolds that have finite fundamental group and contain Klein bottles, Trans. Amer. Math. Soc. 251 (1979), 129-137. MR 531972 (80f:57004)
  • [10] W. Thurston, Geometry and topology of $ 3$-manifolds, Math. Notes, Princeton Univ. Press, Princeton, N. J. (to appear).
  • [11] F. Waldhausen, Gruppen mit Zentrum und $ 3$-dimensionale Mannifaltigkeiten, Topology 6 (1967), 505-517. MR 38 #5223. MR 0236930 (38:5223)
  • [12] -, On irreducible $ 3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56-88. MR 36 #7146. MR 0224099 (36:7146)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57N10, 57N37, 57R95

Retrieve articles in all journals with MSC: 57N10, 57N37, 57R95


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0645327-7
Keywords: Non-Haken $ 3$-manifold, one-sided Heegaard splitting, isotopy class of embeddings, genus $ 3$ nonorientable surface
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society