Characterize triangulable topological spaces

Characterize the class of topological spaces that admit a triangulation (i.e., are homeomorphic to a polyhedron arising from a finite simplicial complex), and ascertain triangulability criteria for non-differentiable manifolds in dimensions greater than three.

Background

Simplicial homology relies on triangulations to translate topological spaces into combinatorial data structures. The text notes that all differentiable manifolds are triangulable and that every topological 2- or 3-manifold admits a triangulation, yet there exist non-smooth 4-manifolds (related to E_8) that do not. Beyond these cases, a comprehensive characterization of triangulable spaces is lacking.

Resolving this would solidify the foundations of computational and combinatorial topology, clarifying when simplicial methods can be applied and guiding algorithmic approaches to homology for general spaces.

References

All differentiable manifolds have triangulations, but a complete characterization of the class of topological spaces that have a triangulation is not known. Every topological 2 or 3-manifold has a triangulation, but there is a (non-smooth) 4-manifold that cannot have a triangulation (it is related to the Lie group E_8 ). The situation for non-differentiable manifolds in higher dimensions remains uncertain.

— Algebraic Topology  (1304.7846 - Robins, 2013) in Subsection "Simplicial complexes" within Section "Homology"