triangulability

Triangulability is the property of a topological space to be homeomorphic to the geometric realization of a simplicial complex. When this holds, the space is said to be triangulable and admits a triangulation, a decomposition into simplices that mirrors its topology. Triangulability thus provides a bridge between continuous spaces and combinatorial models used in geometry and topology.

In the setting of manifolds, triangulability is closely tied to the existence of a piecewise-linear (PL) structure. A triangulable manifold carries a PL structure compatible with its topology, and conversely a PL manifold is triangulable by construction. The precise relationship between triangulations and PL structures depends on dimension. A fundamental obstruction to PL structures, and hence to triangulability in general, is the Kirby–Siebenmann class in cohomology; if this obstruction vanishes, a PL structure (and thus a triangulation) exists.

Results and limitations: every topological manifold of dimension at most three is triangulable. In higher dimensions, triangulability can fail: there exist topological manifolds that do not admit any triangulation due to nonzero obstructions. The Kirby–Siebenmann obstruction provides a concrete criterion for the existence of a PL structure in dimensions where it is applicable.

The Hauptvermutung, a classical conjecture about triangulations, posited that any two triangulations of a manifold have a common subdivision (and hence are equivalent). It is known to hold in low dimensions (notably up to dimension three) but is false in general for higher dimensions.

In summary, triangulability concerns when a space can be modeled by simplicial data; in low dimensions it always holds for manifolds, while in higher dimensions it can fail and is governed by cohomological obstructions related to PL structures.