Summary

Smooth and polyhedral tight immersions exist of all surfaces in all CES, except for:

  • \RP2 in any CES
  • K in two orientable CES

The only remaining open case is \RP2 with one handle in the 3-torus, for which an example seems very unlikely.

Many of the surfaces, including non-orientable ones, can be embedded tightly in some or all CES. Do embeddings exist in the other cases as well?

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