Tightness, torsion, and tubes (Q1202526)

According to \textit{G. Thorbergsson} [Duke Math. J. 57, No. 1, 347-355 (1988; Zbl 0662.53051)], the first nonvanishing integral homology \(H_i (M)\), \(i \geq 1\), of a manifold \(M\) admitting a taut immersion into \(E^n\) has no other torsion than 2-torsion. For the weaker restriction that the manifold admits a tight immersion, the existence of \(p\)-torsion in the homology for \(p \neq 2\) is unknown. All known tight examples with torsion exhibit 2-torsion only. The author presents an example of a 2-dimensional simplicial complex \(M^2\) with 3-torsion in \(H_1 (M^2)\) which admits a polyhedral immersion being \(R_3\)-tight but not \(R_2\)-tight. Furthermore he presents examples of tightly embedded polyhedral manifolds of higher dimension with 3-torsion. These examples are based on the so-called \(2^K\)-construction given by Danzer (see the work of \textit{P. McMullen} and \textit{H. Schulte} [Math. Z. 201, 209-226 (1989; Zbl 0663.51007)]).