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Klein bottles in lens spaces

Hansjörg Geiges and Norman Thies

Vol. 16 (2023), No. 4, 621–636

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 L(4n,2n ± 1) only. We describe four explicit realisations of these embeddings.

Klein bottle, lens space, embedding, Seifert fibration
Mathematical Subject Classification
Primary: 57R40, 57K30, 57M99
Received: 13 May 2022
Accepted: 11 August 2022
Published: 31 October 2023

Communicated by Colin Adams
Hansjörg Geiges
Mathematisches Institut
Universität zu Köln
Norman Thies
Mathematisches Institut
Universität zu Köln