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| {{For|other uses of triangulation in mathematics|Triangulation (disambiguation)}}
| | Andera is what you can call her but she by no means truly liked that title. What me and my family adore is bungee leaping but I've been taking on new things lately. Office supervising is what she does for a living. Some time in the past she chose to live in Alaska and her parents reside close by.<br><br>My site :: tarot readings ([http://www.conocetufe.org/members/profile/116295/GaDickens www.conocetufe.org]) |
| In [[mathematics]], [[topology]] generalizes the notion of [[triangulation (geometry)|triangulation]] in a natural way as follows:
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| A '''triangulation''' of a [[topological space]] <math>X</math> is a [[simplicial complex]] ''K'', homeomorphic to ''X'', together with a [[homeomorphism]] ''h'':''K''<math>\to</math> ''X''.
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| Triangulation is useful in determining the properties of a topological space. For example, one can compute [[Homology (mathematics)|homology]] and [[cohomology]] groups of a triangulated space using simplicial homology and cohomology theories instead of more complicated homology and cohomology theories.
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| ==Piecewise linear structures==
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| {{main|Piecewise linear manifold}}
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| For topological [[manifold]]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, it 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.
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| 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 sphere, a three-dimensional manifold that is not homeomorphic to a sphere.
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| The question of which manifolds have piecewise-linear triangulations has led to much research in topology.
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| [[Differentiable manifold]]s (Stewart Cairns, {{harvs|txt=yes|authorlink=J.H.C. Whitehead|first=J.H.C.|last=Whitehead|year=1940}}, [[Luitzen Egbertus Jan Brouwer|L.E.J. Brouwer]], [[Hans Freudenthal]], {{harvnb|Munkres|1966}}) and [[subanalytic set]]s ([[Heisuke Hironaka]] and Robert Hardt) admit a piecewise-linear triangulation, technically by passing via the [[PDIFF]] category.
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| [[Topological manifold]]s of dimensions 2 and 3 are always triangulable by an [[Hauptvermutung|essentially unique triangulation]] (up to piecewise-linear equivalence); this was proved for [[surface]]s by [[Tibor Radó]] in the 1920s and for [[three-manifold]]s by [[Edwin E. Moise]] and [[R. H. Bing]] in the 1950s, with later simplifications by [[Peter Shalen]] ({{harvnb|Moise|1977}}, {{harvnb|Thurston|1997}}). As shown independently by [[James Munkres]], [[Steve Smale]] and
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| {{harvs|txt=yes|authorlink=J.H.C. Whitehead|first=J.H.C.|last=Whitehead|year=1961}}, each of these manifolds admits
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| a [[smooth structure]], unique up to [[diffeomorphism]] (see {{harvnb|Milnor|2007}}, {{harvnb|Thurston|1997}}).
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| 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 manifold|piecewise-linear]] triangulations (see [[Hauptvermutung]]). On March 13, 2013, [[Ciprian Manolescu]] posted a preprint on the [[arXiv]] claiming to show that there are 5-manifolds, and hence in every dimension greater than or equal to 5, that are not homeomorphic to a simplicial complex, i.e., that do not admit a triangulation.
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| == Explicit methods of triangulation ==
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| 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 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 [[geodesic normal coordinates|normal ball]] with centre ''x''. The interiors of finitely many of the triangles will cover
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| 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.
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| 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''<sup>m</sup>, subdivide a cubical lattice in ''R''<sup>m</sup> into simplices to give a triangulation of ''R''<sup>m</sup>. 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''
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| intersect ''X'' and each ''s''-simplex intersecting ''X''
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| * does so in exactly one interior point;
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| * makes a strictly positive angle with the tangent plane;
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| * lies wholly inside some [[tubular neighbourhood]] of ''X''.
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| These points of intersection and their barycentres (corresponding to higher dimensional simplices intersecting ''X'') generate an ''n''-dimensional simplicial subcomplex in ''R''<sup>m</sup>, lying wholly inside the tubular neighbourhood. The triangulation is given by the projection of this simplicial complex onto ''X''.
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| == Graphs on surfaces ==
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| 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 [[clique graph|clique]]s 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 [[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''<sub>4</sub>.
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| == References ==
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| *{{cite jstor|1968861}}
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| *{{cite jstor|1970286}}
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| *{{citation
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| |authorlink =John Milnor|last=Milnor|first= John W.
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| | title= Collected Works Vol. III, Differential Topology
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| | publisher = American Mathematical Society
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| | year = 2007
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| | isbn = 0-8218-4230-7
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| }}
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| *{{citation
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| | last=Whitney|first= H. |authorlink=Hassler Whitney
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| | title= Geometric integration theory
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| | publisher= Princeton University Press
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| | year =1957
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| |pages =124–135
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| }}
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| *{{citation
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| |authorlink = Jean Dieudonné | last=Dieudonné|first= J.
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| | title= A History of Algebraic and Differential Topology, 1900-1960
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| | publisher= Birkhäuser
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| | year =1989
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| | isbn = 0-8176-3388-X
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| }}
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| *{{citation | last =Jost|first= J. | title= Compact Riemann Surfaces | publisher = Springer-Verlag | year =1997| isbn = 3-540-53334-6}}
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| *{{citation |authorlink=Edwin E. Moise |last = Moise|first= E. |title = Geometric Topology in Dimensions 2 and 3 |publisher= Springer-Verlag| year=1977| isbn=0-387-90220-1}}
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| *{{cite jstor|1970228}}
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| *{{citation | last = Munkres|first= J.|authorlink=James Munkres|title =Elementary Differential Topology, revised edition| series= Annals of Mathematics Studies 54
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| | publisher=Princeton University Press|year=1966|
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| isbn =0-691-09093-9}}
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| *{{citation | last = Thurston|first= W. |authorlink=William Thurston|title= Three-Dimensional Geometry and Topology, Vol. I| publisher = Princeton University Press
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| | year =1997| isbn= 0-691-08304-5}}
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| *{{citation
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| | last = Hartsfeld|first= N.|author2-link=Gerhard Ringel|last2=Ringel|first2= G.
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| | title = Clean triangulations
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| | journal = Combinatorica
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| | volume = 11
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| | year = 1991
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| | pages = 145–155
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| | doi = 10.1007/BF01206358
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| | issue = 2}}
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| *{{citation
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| | last= Larrión|first= F.|last2= Neumann-Lara|first2= V.|last3= Pizaña|first3= M. A.
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| | title = Whitney triangulations, local girth and iterated clique graphs
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| | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
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| | volume = 258
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| | year = 2002
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| | pages = 123–135
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| | url = http://xamanek.izt.uam.mx/map/papers/cuello10_DM.ps
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| | doi= 10.1016/S0012-365X(02)00266-2}}
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| *{{citation
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| | last = Malnič|first= Aleksander|last2= Mohar|first2= Bojan|author2-link=Bojan Mohar
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| | title = Generating locally cyclic triangulations of surfaces
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| | journal = Journal of Combinatorial Theory, Series B
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| | volume = 56
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| | issue = 2
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| | year = 1992
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| | pages = 147–164
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| | doi = 10.1016/0095-8956(92)90015-P}}
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| [[Category:Topology]]
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| [[Category:Algebraic topology]]
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| [[Category:Geometric topology]]
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| [[Category:Structures on manifolds]]
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| [[Category:Triangulation (geometry)]]
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Andera is what you can call her but she by no means truly liked that title. What me and my family adore is bungee leaping but I've been taking on new things lately. Office supervising is what she does for a living. Some time in the past she chose to live in Alaska and her parents reside close by.
My site :: tarot readings (www.conocetufe.org)