Operations on 3-dimensional small covers (Q606354)

A small cover, which was introduced and studied by \textit{M. W. Davis} and \textit{T. Januszkiewicz} in the paper [Duke Math. J. 62, No.2, 417--451 (1991; Zbl 0733.52006)], is a smooth closed manifold with an effective action of a 2-torus such that the action is locally standard and the orbit space of the action is a simple convex polytope. Each small cover can be reconstructed from a simple convex polytope with a characteristic function. This gives a possibility for the topological classification of small covers by carrying out the equivariant surgeries on the polytopes. Indeed, \textit{I. V. Izmest'ev} [Math. Notes 69, No. 3, 340--346 (2001); translation from Mat. Zametki 69, No. 3, 375--382 (2001; Zbl 0991.57016)] obtained a classification result for a kind of special 3-dimensional small covers (called also linear models) by using two operations of the equivariant connected sum \(\sharp\) and the equivariant Dehn surgery \(\natural\). Nishimura [Equivariant surgeries of small covers (in Japanese), RIMS Kokyuroku, 1393, 44--47 (2004)] extended the above case to that of oriented 3-dimensional small covers, and also obtained a classification result by using three operations \(\sharp\), \(\natural\) and \(\flat\). In the paper [\textit{Z. Lü} and \textit{L. Yu}, Forum Math. 23, No. 2, 245--284 (2011; Zbl 1222.52015)], all 3-dimensional small covers in the general case were classified topologically by using six operations \(\sharp, \sharp^e, \sharp^{\text{eve}}\), \(\natural\), \(\sharp^\triangle\), \(\sharp^C\) on five basic small covers. The author of the paper under review investigates the relations among three classification results, especially the relations among the associated operations. The main result of the paper is as follows: Theorem. The operations \(\flat\), \(\sharp^e\), \(\sharp^{\text{eve}}\) can be obtained by using \(\sharp\), \(\natural\), and the operations \(\sharp^\triangle\), \(\sharp^C\) can be obtained by using \(\sharp\), \(\natural\) and \(\natural_0\), where \(\natural_0\) is a new operation introduced by the author of the paper under review.