Nonorientable surfaces in some non-Haken 3-manifolds

Rubinstein, J. H.

doi:10.1090/S0002-9947-1982-0645327-7

Nonorientable surfaces in some non-Haken $3$-manifolds
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by J. H. Rubinstein

Trans. Amer. Math. Soc. 270 (1982), 503-524

DOI: https://doi.org/10.1090/S0002-9947-1982-0645327-7

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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.

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