The paper studies the connective complex K-theoretic Euler class of a certain bundle associated to a Euclidean immersion of the lens space L(2k)2n+1> to show that this manifold cannot be immersed in ℝ4n−2α(n) if k [ges ] α(n). The non-immersion is best possible in many cases. This suggests a close relationship between the immersion problem for complex projective spaces and that for ‘high’ 2-torsion lens spaces.