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mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
,
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are p ...
generalizes the notion of triangulation in a natural way as follows: :A triangulation of a
topological space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
''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
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics, a manifold is a topological space that locally resembles Euclidean space ...
s, 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 graph, 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 (topology), suspension of the Poincaré homology sphere, 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 homology sphere, Poincaré sphere, a three-dimensional manifold that is not homeomorphic to a sphere. This is the double suspension theorem, due to James W. Cannon and R.D. Edwards in the 1970s.J. W. Cannon
''Shrinking cell-like decompositions of manifolds. Codimension three.''
Annals of Mathematics (2), 110 (1979), no. 1, 83–112.
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), 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 Hauptvermutung, essentially unique triangulation (up to piecewise-linear equivalence); this was proved for Surface (topology), 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. As shown independently by James Munkres, Steve Smale and J. H. C. Whitehead, each of these manifolds admits a smooth structure, unique up to diffeomorphism. 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, Robion Kirby, Rob Kirby and Larry Siebenmann constructed manifolds that do not have piecewise linear manifold, 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.

# Explicit methods of triangulation

An important special case of topological triangulation is that of two-dimensional surfaces, or Manifold, closed 2-manifolds. There is a standard proof that smooth compact surfaces can be triangulated. Indeed, if the surface is given a Riemannian metric, each point ''x'' is contained inside a small convex geodesic triangle lying inside a geodesic normal coordinates, 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 Whitney embedding theorem, 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 (mathematics), 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 (topology), surface is an embedding of a Graph (discrete mathematics), graph onto the surface in such a way that the faces of the embedding are exactly the clique (graph theory), cliques of the graph. 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-Skeleton (topology), skeletons of Whitney triangulations are exactly the Neighbourhood (graph theory), locally cyclic graphs other than ''K''4.