All orientable 2-manifolds have finitely many minimal triangulations
The boundary of a tetrahedron is a triangulation of the 2-sphere \(S_ 2\) that is minimal in the sense that no edge can be shrunk to a single vertex. A classical result of Steinitz [\textit{E. Steinitz}, Vorlesungen über die Theorie der Polyeder under Einschluß der Elemente der Topologie. Aus dem Nachlaß hrsg. u. erg. v. Hans Rademacher (1934; Zbl 0009.36503)] asserts that every triangulation of \(S_ 2\) can be reduced to this one by shrinking edges, i.e. there are no other minimal triangulations of \(S_ 2\). The number of minimal triangulations of the projective plane and the torus are known to be 2 and 22, respectively. The authors show that this number is finite for every compact orientable 2-manifold. The proof is by induction on the genus, using the following ideas. Every edge of a minimal triangulation of \(M\neq S_ 2\) belongs to a 3-circuit that does not bound a cell, and the number of such circuits passing through a given vertex is bounded by a constant depending only on the genus g of M. This allows to produce from any minimal triangulation of \(M_ g\) a minimal triangulation of \(M_{g-1}\) using surgery and a bounded number of edge shrinkings.
scientific article; zbMATH DE number 4067881