Triangulation (topology)

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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.

For topological manifolds, there is a slightly stronger notion of triangulation: a piecewise-linear triangulation (sometimes just called a triangulation) is a triangulation with the extra property that the link of any simplex is a piecewise-linear sphere. For manifolds of dimension at most 4 this extra property automatically holds, but in dimension n ≥ 5 the (n−3)-fold suspension of the Poincaré sphere is a topological manifold (homeomorphic to the n-sphere) with a triangulation that is not piecewise-linear: it has a simplex whose link is the suspension of the Poincaré sphere, which is not a manifold (though it is a homology manifold).

Differentiable manifolds (Stewart Cairns, J.H.C. Whitehead (1940), L.E.J. Brouwer, Hans Freudenthal) and subanalytic sets (Heisuke Hironaka and Robert Hardt) admit a piecewise-linear triangulation.

Topological manifolds of dimensions 2 and 3 are always triangulable by an essentially unique triangulation (up to piecewise-linear equivalence); this was proved for surfaces by Tibor Radó in the 1920s and for three-manifolds by Edwin Moise and RH Bing in the 1950s, with later simplifications by Peter Shalen (Moise 1977, Thurston 1997). As shown independently by James Munkres, Steve Smale and J.H.C. Whitehead (1961), each of these manifolds admits a smooth structure, unique up to diffeomorphism (see Munkres 1966, Milnor 2007, Thurston 1997).

In dimension 4, however, the E8 manifold does not admit a triangulation, and some compact 4-manifolds have an infinite number of triangulations, all piecewise-linear inequivalent. In dimension greater than 4, the question of whether all topological manifolds have triangulations is an open problem, though it is known that some do not have piecewise-linear triangulations (see Hauptvermutung).

[edit] Explicit methods of triangulation

An important special case of topological triangulation is that of two-dimensional surfaces, or closed 2-manifolds. There is a standard proof that smooth closed surfaces can be triangulated (see Jost 1997). Indeed, if the surface is given a Riemannian metric, each point x is contained inside a small convex geodesic triangle lying inside a normal ball with centre x. The interiors of finitely many of the triangles will cover the surface; since edges of different triangles either coincide or intersect transversally, this finite set of triangles can be used iteratively to construct a triangulation.

Another simple procedure for triangulating differentiable manifolds was given by Hassler Whitney in 1957, based on his embedding theorem. In fact, if X is a closed n-submanifold of Rm, subdivide a cubical lattice in Rm into simplices to give a triangulation of Rm. By taking the mesh of the lattice small enough and slightly moving finitely many of the vertices, the triangulation will be in general position with respect to X: thus no simplices of dimension < s=m-n intersect X and each s-simplex intersecting X

  • does so in exactly one interior point;
  • makes a strictly positive angle with the tangent plane;
  • lies wholly inside some tubular neighbourhood of X.

These points of intersection and their barycentres (corresponding to higher dimensional simplices intersecting X) generate an n-dimensional simplicial subcomplex in Rm, lying wholly inside the tubular neighbourhood. The triangulation is given by the projection of this simplicial complex onto X.

[edit] Graphs on surfaces

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 Gerhard 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.

Plantri and Fullgen are programs for generation of certain types of planar graphs; they were developed by Gunnar Brinkmann and Brendan McKay.[1]

[edit] References

  • Whitehead, J.H.C. (1940), “On C¹ complexes”, Ann. of Math. 41: 809-824 
  • Whitehead, J.H.C. (1961), “Manifolds with tranverse fields in Euclidean space”, Ann. of Math. 73: 154-212 
  • Milnor, John W. (2007), Collected Works Vol. III, Differential Topology, American Mathematical Society, ISBN 0821842307 
  • Whitney, H. (1957), Geometric integration theory, Princeton University Press, pp. 124-135 
  • Dieudonné, J. (1989), A History of Algebraic and Differential Topology, 1900-1960, Birkhäuser, ISBN 081763388X 
  • Jost, J. (1997), Compact Riemann Surfaces, Springer-Verlag, ISBN 3-540-53334-6 
  • Moise, E. (1977), Geometric Topology in Dimensions 2 and 3, Springer-Verlag, ISBN 0387902201 
  • Munkres, J. (1966), Elementary Differential Topology, revised edition, Annals of Mathematics Studies 54, Princeton University Press, ISBN 0691090939 
  • Thurston, W. (1997), Three-Dimensional Geometry and Topology, Vol. I, Princeton University Press, ISBN 0-691-08304-5 
  • Hartsfeld, N. & Ringel, G. (1991), “Clean triangulations”, Combinatorica 11: 145–155