A tight polyhedral immersion of the twisted surface of Euler characteristic \(-3\) (Q5939201)

A map \(f:M\to \mathbb{R}^3\) is called tight provided that the preimage of every half-space of \(\mathbb{R}^3\) is connected, that is every plane cuts \(f(M)\) into at most two pieces. Tight immersions of surfaces are the subject of the paper. The Klein bottle and the projective plane can not be tightly immersed into \(\mathbb{R}^3\), while all other surfaces except the projective plane with one handle can. The existence of tight immersions may depend on the considered category. Such an example is the projective plane with one handle which does not admit smooth tight immersion but admits a polyhedral one. For all surfaces but Klein bottle and projective plane there exist tight polyhedral immersions. Since the immersions are classified by the image homotopy equivalence relation one may consider the restriction of this classification to tight immersions. The restriction will give the classification of tight immersions if every class of immersions can be represented by a tight one whenever tight immersions are possible. This task was completed up to three classes in the author's previous paper [ibid. 35, No. 4, 863-873 (1996; Zbl 0858.53051)]. The present paper proves that two of these three classes can be represented by a polyhedral tight immersion while the third class still remains open.