Klein bottles in lens spaces

Geiges, Hansjörg; Thies, Norman

doi:10.2140/involve.2023.16.621

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Klein bottles in lens spaces

Hansjörg Geiges and Norman Thies

Vol. 16 (2023), No. 4, 621–636

DOI: 10.2140/involve.2023.16.621

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

Keywords

Klein bottle, lens space, embedding, Seifert fibration

Mathematical Subject Classification

Primary: 57R40, 57K30, 57M99

Milestones

Received: 13 May 2022

Accepted: 11 August 2022

Published: 31 October 2023

Communicated by Colin Adams

Authors

Hansjörg Geiges
Mathematisches Institut Universität zu Köln Köln Germany

Norman Thies
Mathematisches Institut Universität zu Köln Köln Germany

Vol. 16, No. 4, 2023