Some 3-manifolds which admit Klein bottles

Kim, Paik Kee

doi:10.1090/S0002-9947-1978-0506621-2

Some $3$-manifolds which admit Klein bottles
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by Paik Kee Kim

Trans. Amer. Math. Soc. 244 (1978), 299-312

DOI: https://doi.org/10.1090/S0002-9947-1978-0506621-2

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Abstract:

Consider a closed, orientable, irreducible 3-manifold M with $\left | {{\pi _1}(M)} \right | < \infty$, in which a Klein bottle can be embedded. We present a classification of the spaces M and show that, if ${\pi _1}(M)$ is cyclic, then M is homeomorphic to a lens space. Note that all surfaces of even genus can be embedded in each space M. We also classify all free involutions on lens spaces whose orbit spaces contain Klein bottles.

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