|
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== (詳細は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, 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. 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. The question of which manifolds have piecewise-linear triangulations has led to much research in topology. Differentiable manifolds (Stewart Cairns, , L.E.J. Brouwer, Hans Freudenthal, ) 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 (, ). As shown independently by James Munkres, Steve Smale and , each of these manifolds admits a smooth structure, unique up to diffeomorphism (see , ). 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 (). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Triangulation (topology)」の詳細全文を読む スポンサード リンク
|