No Tight Klein Bottle

If $\alpha(f)$ is the number of top cycles for a tight immersion of $M$ (not a sphere), then

2\le\alpha(f)\le2-\chi(M).

 For the Klein bottle: $\chi=0$ so $\alpha(f)=2$, $M^+$ is a cylinder, $M^-$ is a cylinder that reverses the orientation of its boundary, a contradiction.