All 2-manifolds have finitely many minimal triangulations

Link Conditions for Simplifying Meshes with Embedded Structures

“…Early work on minimum and minimal triangulations studied the smallest possible mesh that can be reached without violating the topology [15], [16]. Different from topology preserving simplification, controlled topology simplification helps to remove topological noises like small holes, while retaining important topological characteristics of the mesh.…”

Recognizing Shrinkable Complexes Is NP-Complete

“…An admissible simplex collapse (called elementary collapse in [23]) is the operation of removing a simplex and one of its faces if this face belongs to no other simplex. 4 Such collapses preserve the homotopy type. Similarly to edge contractions, collapses are often used to simplify simplicial complexes, and a simplicial complex is said collapsible if it can be reduced to a single vertex by a sequence of admissible collapses.…”

“…They use this property to prove that a compact 2-manifold (orientable or not) of fixed genus admits finitely many triangulations that are (shrink) irreducible [3,4]. For instance, the number of irreducible triangulations of the torus is 21 [17] and it is at most 396 784 for the double torus [22].…”

A Note on the Complexity of Real Algebraic Hypersurfaces

“…An often discussed concept in this context is an irreducible triangulation of a 2-manifold, i.e., a triangulation where no edge can be contracted without changing the topology. It has been shown that only finitely many irreducible triangulations exist [2], and they have been enumerated explicitly for the torus [11]. Although these results aim in a somewhat similar direction, algebraic surfaces are in general not 2-manifolds and need different techniques to be analyzed.…”