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Non-orientable surfaces in 3-manifolds. (English) Zbl 0763.57012

If \(i:F\to A\) is a proper embedding of a non-orientable surface \(F\) in a 3-manifold \(A\) then the Stiefel-Whitney class \(w_ 1(F)\) lies in the image of \(i^*:H^ 1(A;\mathbb{Z}_ 2)\to H^ 1(F;\mathbb{Z}_ 2)\). This and related results are used to study embeddings of non-orientable surfaces in 3-manifolds. For example it is shown that \(U_ g\) embeds in \(\mathbb{R} P^ 3\) if and only if \(g\) is odd, and \(U_ g\) embeds in \(S^ 2\times S^ 1\) if and only if \(g\) is even and that \(U_ 2\), the Klein bottle, embeds in \(M\times S^ 1\) (where \(M\) is an orientable surface) if and only if \(M\approx S^ 2\); here \(U_ g\) is the closed, connected, non- orientable surface of genus \(g\). Embeddability of \(U_ g\) in the lens spaces is also studied.

MSC:

57N35 Embeddings and immersions in topological manifolds
57R20 Characteristic classes and numbers in differential topology
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
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References:

[1] F.Apéry, Models of the Projective Plane. Braunschweig 1987.
[2] G. E. Bredon andJ. W. Wood, Non orientable surfaces in orientable 3-Manifolds. Invent. Math.7, 83-110 (1969). · Zbl 0175.20504 · doi:10.1007/BF01389793
[3] G.Burde and H.Zieschang, Knots. Berlin 1985.
[4] M. M.Cohen, A Course in Simple Homotopy Theory. Graduate Texts in Math. Berlin-Heidelberg-New York 1973. · Zbl 0261.57009
[5] W.End, Homotopy Types of 2-Manifolds. To appear. · Zbl 0443.55008
[6] D. B. A. Epstein, Curves on 2-Manifolds and Isotopies. Acta Math.115, 83-107 (1966). · Zbl 0136.44605 · doi:10.1007/BF02392203
[7] J. Hass andJ. Hughes, Immersions of surfaces in 3-manifolds. Topology24, 97-112 (1985). · Zbl 0527.57020
[8] J.Hempel, 3-Manifolds. Ann. of Math. Stud.86, Princeton 1976. · Zbl 0345.57001
[9] M.Hirsch, Differential Topology. Berlin-Heidelberg-New York 1976.
[10] W. Jaco, Surfaces embedded inM 2{\(\times\)}S 1. Canad. J. Math.22, 335-346 (1970). · Zbl 0205.53501 · doi:10.4153/CJM-1970-063-x
[11] J. W.Milnor and D.Husemoller, Symmetric Bilinear Forms. Berlin-Heidelberg-New York 1973. · Zbl 0292.10016
[12] J. W.Milnor and J.Stasheff, Characteristic classes. Ann. of Math. Stud.76, Princeton 1974. · Zbl 0298.57008
[13] D.Rolfsen, Knots and Links. Berkley 1976.
[14] H.Seifert and W.Threlfall, Topologie. New York 1934.
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