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.

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