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Encyclopedia > Obstruction

In mathematics, obstruction theory is a name for more than one mathematical theory. Euclid, detail from The School of Athens by Raphael. ... In mathematics, theory is used informally to refer to a body of knowledge about mathematics. ...

The older meaning in homotopy theory relates to a procedure, inductive with respect to dimension, for extending a continuous mapping defined on a simplicial complex, or CW complex. It involves cohomology groups with coefficients in homotopy groups to define obstructions to extensions. For example, with a mapping from a simplicial complex X to another, Y, defined initially on the 0-skeleton of X (the vertices of X), an extension to the 1-skeleton will be possible whenever Y is sufficiently path-connected. Extending from the 1-skeleton to the 2-skeleton means filling in the images of the solid triangles from X, given the image of the edges. An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ... In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...

Another theory of the same name in geometric topology is concerned with when a topological manifold has a piecewise linear structure, and when a piecewise linear manifold has a differentiable structure. In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ... A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ... In mathematics, a structure on a set is some additional mathematical objects that, loosely speaking, attach to the set, making it easier to visualize or work with. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space. ...

In dimension at most 2 (Rado), and 3 (Moise), the notions of topological manifolds and piecewise linear manifolds coincide. In dimension 4 they are not the same. Moise (d. ...

In dimensions at most 3 the notions of piecewise linear manifolds and differentiable manifolds coincide. In dimension 4 they are not the same.

See also

  • Kirby-Siebenmann class

  Results from FactBites:
MedlinePlus Medical Encyclopedia: Intestinal obstruction (581 words)
Intestinal obstruction involves a partial or complete blockage of the bowel that results in the failure of the intestinal contents to pass through.
Obstruction of the bowel may be caused by ileus -- in which the bowel doesn't function correctly but there is no "mechanical" (anatomic) problem -- or by mechanical causes.
Surgery to relieve the obstruction may be necessary if decompression by NG tube does not relieve the symptoms, or if tissue death is suspected.
  More results at FactBites »



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