Klein bottles in lens spaces

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