Klein bottles in lens spaces

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.