Klein bottles in lens spaces
Hansj\"org Geiges, Norman Thies

TL;DR
This paper proves that Klein bottles embed only in specific lens spaces, providing explicit constructions and completing the understanding of nonorientable surface embeddings in these manifolds.
Contribution
It offers a direct proof identifying exactly which lens spaces admit Klein bottle embeddings and provides explicit embedding examples.
Findings
Klein bottles embed only in lens spaces of the form L(4n, 2n±1).
Four explicit realizations of Klein bottle embeddings are constructed.
The result completes the classification of Klein bottle embeddings in lens spaces.
Abstract
Bredon and Wood have given a complete answer to the embeddability question for nonorientable surfaces in lens spaces. They formulate their result in terms of a recursive formula that determines, for a given lens space, the minimal genus of embeddable nonorientable surfaces. Here we give a direct proof that, amongst lens spaces as target manifolds, the Klein bottle embeds into only. We describe four explicit realisations of these embeddings.
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Taxonomy
TopicsGeometric and Algebraic Topology · Mathematical Dynamics and Fractals · Homotopy and Cohomology in Algebraic Topology
