Ikketriangulerbare

Ikketriangulerbare refers to topological manifolds that do not admit any triangulation. In other words, such a manifold cannot be given a simplicial complex structure whose underlying space is homeomorphic to the manifold. Triangulability is equivalent to the existence of a piecewise-linear (PL) structure on the manifold, so a non-triangulable manifold lacks any PL structure as well.

A key tool in the study of triangulability is obstruction theory. For topological manifolds of dimension five and higher, the Kirby–Siebenmann obstruction KS(M) lies in the cohomology group H^4(M; Z2). If KS(M) is nonzero, M does not admit a PL structure, and therefore cannot be triangulated. Conversely, KS(M) = 0 implies the existence of a PL structure and hence a triangulation. This obstruction provides a complete criterion for triangulability in the high-dimensional setting.

Historical and practical significance: it is known that nontriangulable manifolds exist in dimensions five and higher. The existence results were developed in the 1960s and 1970s, with contributions by C. T. C. Wall and the subsequent refinement of obstruction theory by Kirby and Siebenmann. These results show that triangulation, while automatic in low dimensions, fails in general in higher dimensions, marking a fundamental boundary in the relationship between topology and combinatorial structures.

In low dimensions (dimension up to three) every topological manifold is triangulable, reflecting a simpler landscape for triangulations in those cases. The four-dimensional situation is more subtle and does not fit the same general obstruction framework as higher dimensions.