Nonorientable surfaces in some nonHaken manifolds
Author:
J. H. Rubinstein
Journal:
Trans. Amer. Math. Soc. 270 (1982), 503524
MSC:
Primary 57N10; Secondary 57N37, 57R95
MathSciNet review:
645327
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Abstract: If a closed, irreducible, orientable manifold does not possess any sided incompressible surfaces, then it can be very useful to investigate embedded onesided surfaces in of minimal genus. In this paper such manifolds are studied which admit embeddings of the nonorientable surface of genus . We prove that a manifold of the above type has at most different isotopy classes of embeddings of representing a fixed element of . If is either a binary octahedral space, an appropriate lens space or Seifert manifold, or if has a particular type of fibered knot, then it is shown that the embedding of in realizing a specific homology class is unique up to isotopy.
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 Joan S. Birman and D. Chillingworth, On the homeotopy group of a nonorientable surface, Proc. Cambridge Philos. Soc. 71 (1972), 437448. MR 45 #9334. MR 0300288 (45:9334)
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 Joan S. Birman and J. H. Rubinstein, Homeotopy groups of some nonHaken manifolds (to appear).
 [3]
 Joan S. Birman, F. GonzalesAcuna and J. M. Montesinos, Heegaard splittings of prime manifolds are not unique, Michigan Math. J. 23 (1976), 97103. MR 55 #4177. MR 0431175 (55:4177)
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 G. Bredon and J. Wood, Nonorientable surfaces in orientable manifolds, Invent. Math. 7 (1969), 83110. MR 39 #7616. MR 0246312 (39:7616)
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 M. Culler, W. Jaco and J. H. Rubinstein, J. London Math. Soc. (to appear).
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 J. Milnor, Lectures on the cobordism theorem, Math. Notes, Princeton Univ. Press, Princeton, N.J., 1965. MR 32 #8352. MR 0190942 (32:8352)
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 P. Orlik, Seifert manifolds, Lecture Notes in Math., vol. 291, SpringerVerlag, Berlin and New York, 1972. MR 54 #13950. MR 0426001 (54:13950)
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 J. H. Rubinstein, Onesided Heegaard splittings of manifolds, Pacific J. Math. 76 (1978), 185200. MR 0488064 (58:7635)
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 , On manifolds that have finite fundamental group and contain Klein bottles, Trans. Amer. Math. Soc. 251 (1979), 129137. MR 531972 (80f:57004)
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 , On irreducible manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 5688. MR 36 #7146. MR 0224099 (36:7146)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198206453277
PII:
S 00029947(1982)06453277
Keywords:
NonHaken manifold,
onesided Heegaard splitting,
isotopy class of embeddings,
genus nonorientable surface
Article copyright:
© Copyright 1982
American Mathematical Society
