Talk Slides (2007-04-30.ualbany) Tight: No Tight Klein Bottle

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.