We have only the projective plane with handles left to consider.
Otto used my model of the tight projective plane with one handle to construct tight immersions of $\RP2$ with two handles in every CES. The idea is to remove the outer handle and use the central core as though it were a "handle" being added to a tight "tube" torus. (Instead of adding the core to a sphere with two disks removed, it is added to a torus with two disks removed, so has one additional handle.) |