No Tight Projective Plane
For $M$ not a sphere, there must be at least one top cycle.
Recall that a top cycle:
 is a nontrivial 1cycle, and
 has an orientable neighborhood.
But:
 $\RP2$ has only one nontrivial homology class.
 It's neighborhood is nonorientable.
 So there can be no tight $\RP2$.



