# Triangulation (topology)

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*→*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.

## Piecewise linear structures

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 – defined for dimensions 0, 1, 2, . . . inductively – that the link of any simplex is a piecewise-linear sphere. The *link* of a simplex *s* in a simplicial complex *K* is a subcomplex of *K* consisting of the simplices *t* that are disjoint from *s* and such that both *s* and *t* are faces of some higher-dimensional simplex in *K*. For instance, in a two-dimensional piecewise-linear manifold formed by a set of vertices, edges, and triangles, the link of a vertex *s* consists of the cycle of vertices and edges surrounding *s*: if *t* is a vertex in this cycle, *t* and *s* are both endpoints of an edge of *K*, and if *t* is an edge in this cycle, it and *s* are both faces of a triangle of *K*. This cycle is homeomorphic to a circle, which is a 1-dimensional sphere. But in this article the word "triangulation" is just used to mean homeomorphic to a simplicial complex.

For manifolds of dimension at most 4, any triangulation of a manifold is a piecewise linear triangulation: In any simplicial complex homeomorphic to a manifold, the link of any simplex can only be homeomorphic to a sphere. 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 Poincaré sphere, a three-dimensional manifold that is not homeomorphic to a sphere. This is the double suspension theorem, due to R.D. Edwards in the 1970s.[1][2][3]

The question of which manifolds have piecewise-linear triangulations has led to much research in topology. Differentiable manifolds (Stewart Cairns, J. H. C. Whitehead, L. E. J. Brouwer, Hans Freudenthal, James Munkres),[4][5] and subanalytic sets (Heisuke Hironaka and Robert Hardt) admit a piecewise-linear triangulation, technically by passing via the PDIFF category. 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 E. Moise and R. H. Bing in the 1950s, with later simplifications by Peter Shalen.[6][7] As shown independently by James Munkres, Steve Smale and J. H. C. Whitehead,[8][9] each of these manifolds admits a smooth structure, unique up to diffeomorphism.[7][10] 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, Rob Kirby and Larry Siebenmann constructed manifolds that do not have piecewise-linear triangulations (see Hauptvermutung). Further, Ciprian Manolescu proved that there exist compact manifolds of dimension 5 (and hence of every dimension greater than 5) that are not homeomorphic to a simplicial complex, i.e., that do not admit a triangulation.[11]

## 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.[12] 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,[13] based on his embedding theorem. In fact, if *X* is a closed *n*-submanifold of *R*^{m}, subdivide a cubical lattice in *R*^{m} into simplices to give a triangulation of *R*^{m}. 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 *R*^{m}, lying wholly inside the tubular neighbourhood. The triangulation is given by the projection of this simplicial complex onto *X*.

## 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.[14][15][16] Equivalently, every face is a triangle, every triangle is a face, and the graph is not itself a clique. The clique complex of the graph is then homeomorphic to the surface. The 1-skeletons of Whitney triangulations are exactly the locally cyclic graphs other than *K*_{4}.

## References

- Edwards, Robert D. (2006),
*Suspensions of Homology Spheres*, arXiv:math/0610573 (*reprint of private, unpublished manuscripts from the 1970's*) - Edwards, R. D. (1980), "The topology of manifolds and cell-like maps", in Lehto, O. (ed.),
*Proceedings of the International Congress of Mathematicians, Helsinki, 1978*, Acad. Sci. Fenn, pp. 111–127 - Cannon, J. W. (1978), "Σ
^{2}H^{3}= S^{5}/ G",*Rocky Mountain J. Math.*,**8**: 527–532 - Whitehead, J. H. C. (October 1940), "On
*C*^{1}-Complexes",*Annals of Mathematics*, Second Series,**41**(4): 809–824, doi:10.2307/1968861, JSTOR 1968861 - Munkres, James (1966),
*Elementary Differential Topology, revised edition*, Annals of Mathematics Studies 54, Princeton University Press, ISBN 0-691-09093-9 - Moise, Edwin (1977),
*Geometric Topology in Dimensions 2 and 3*, Springer-Verlag, ISBN 0-387-90220-1 - Thurston, William (1997),
*Three-Dimensional Geometry and Topology, Vol. I*, Princeton University Press, ISBN 0-691-08304-5 - Munkres, James (1960), "Obstructions to the smoothing of piecewise-differentiable homeomorphisms",
*Annals of Mathematics*,**72**(3): 521–554, doi:10.2307/1970228, JSTOR 1970228 - Whitehead, J.H.C. (1961), "Manifolds with Transverse Fields in Euclidean Space",
*The Annals of Mathematics*,**73**(1): 154–212, doi:10.2307/1970286, JSTOR 1970286 - Milnor, John W. (2007),
*Collected Works Vol. III, Differential Topology*, American Mathematical Society, ISBN 0-8218-4230-7 - Manolescu, Ciprian (2016), "Pin(2)-equivariant Seiberg–Witten Floer homology and the Triangulation Conjecture",
*J. Amer. Math. Soc.*,**29**: 147–176, arXiv:1303.2354, doi:10.1090/jams829 - Jost, Jürgen (1997),
*Compact Riemann Surfaces*, Springer-Verlag, ISBN 3-540-53334-6 - Whitney, Hassler (1957),
*Geometric integration theory*, Princeton University Press, pp. 124–135 - Hartsfeld, N.; Ringel, G. (1991), "Clean triangulations",
*Combinatorica*,**11**(2): 145–155, doi:10.1007/BF01206358 - Larrión, F.; Neumann-Lara, V.; Pizaña, M. A. (2002), "Whitney triangulations, local girth and iterated clique graphs",
*Discrete Mathematics*,**258**: 123–135, doi:10.1016/S0012-365X(02)00266-2 - 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

## Further reading

- Dieudonné, Jean (1989),
*A History of Algebraic and Differential Topology, 1900–1960*, Birkhäuser, ISBN 0-8176-3388-X