Non-orientable surfaces in 3-manifolds
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.
proper embeddings of a non-orientable surface in a 3-manifold
scientific article; zbMATH DE number 119848