Tight immersions of simplicial surfaces in three space (Q1922750)

The existence or non-existence of a tight immersion of the closed surface with \(\chi = -1\) into 3-space has been an open problem for more than 30 years. \textit{F. Haab} [Comment. Math. Helv. 67, No. 2, 182-202 (1992; Zbl 0763.53060)] proved that there is no smooth tight immersion of this kind. Recently, the author obtained a polyhedral example, see his forthcoming paper entitled `Tightness for smooth and polyhedral immersions of the real projective plane with one handle'. \textit{U. Pinkall} [Topology 25, 475-481 (1986; Zbl 0605.53027)] gave examples of tight immersions in non-standard homotopy classes (image homotopy classes in the polyhedral case). In this remarkable paper, the author improves Pinkall's result by giving tight simplicial examples in all but two of the immersion classes under image homotopy for which tight immersions are possible, i.e., not known to be impossible. In particular, he points out an error in one of Pinkall's key examples (which is non-orientable rather than orientable) and resolves this problem by another example. The author's key example here is a new tight simplicial immersion of the twisted torus with two standard handles. The two remaining open cases are the twisted torus with a standard handle and the connected sum of the twisted Klein bottle with a Boy surface and with an extra standard handle.