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.