Otto's Conjectures

Otto conjectures that no smooth tight immersions of \RP2 with two handles exist in any CES, and that no tight immersion (smooth or polyhedral) of \RP2 with one handle exists in any CES.

It turns out that both these conjectures are false.

Indeed there exist embeddings of \RP2 with one handle in several CES, and immersions in all the others except one.

This gives us smooth embeddings or immersions of \RP2 with two handles in all CES but one, by the usual addition of handles. As it happens, the final CES also allows a smooth immersion of this surface.