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