Triangulation (topology)

From Wikipedia, the free encyclopedia

For other uses of triangulation in mathematics, see Triangulation (disambiguation).

In mathematics, topology generalizes the notion of triangulation in a natural way as follows:

A triangulation of a topological space $X$ is a simplicial complex $K$ , homeomorphic to $X$ , together with a homeomorphism $h:K\to X$ .

Triangulation is useful in determining the properties of a topological space. For example, one can compute homology and cohomology groups of a triangulated space using simplicial homology and cohomology theories instead of more complicated homology and cohomology theories.

It is known that subanalytic sets and differentiable manifolds admit a triangulation. However, some topological manifolds do not admit a triangulation (see Hauptvermutung).

[edit] Triangulation of surfaces

An important special case of topological triangulation is that of two-dimensional surfaces, or closed 2-manifolds.

A Whitney triangulation or clean triangulation of a surface is an embedding of a graph onto the surface in such a way that the faces of the embedding are exactly the cliques of the graph (Hartsfeld and Ringel 1981; Larrión et al 2002; Malnič and Mohar 1992). Equivalently, every face is a triangle, every triangle is a face, and the graph is not itself a clique. The 1-skeletons of Whitney triangulations are exactly the locally cyclic graphs other than K₄.

[edit] References

Hartsfeld, N.; Ringel, G. (1991). "Clean triangulations". Combinatorica 11: 145–155.

Larrión, F.; Neumann-Lara, V.; Pizaña, M. A. (2002). "Whitney triangulations, local girth and iterated clique graphs". Discrete Mathematics 258: 123–135.

Malnič, Aleksander; Mohar, Bojan (1992). "Generating locally cyclic triangulations of surfaces". Journal of Combinatorial Theory, Series B 56 (2): 147–164. DOI:10.1016/0095-8956(92)90015-P.