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EspañolOBSTRUCTION THEORY
In mathematics, 'obstruction theory' is a name given to two different mathematical theories:
The older meaning for obstruction theory 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.
In geometric topology, obstruction theory is concerned with when a topological manifold has a piecewise linear structure, and when a piecewise linear manifold has a differentiable structure.
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.
In dimensions at most 6 the notions of piecewise linear manifolds and differentiable manifolds coincide.
★ Kirby-Siebenmann class
★ The wild world of 4-manifolds, , Alexandru, Scorpan, American Mathematical Society, 2005, ISBN 0-8218-3749-4
| Contents |
| In homotopy theory |
| In geometric topology |
| See also |
| Reference |
In homotopy theory
The older meaning for obstruction theory 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.
In geometric topology
In geometric topology, obstruction theory is concerned with when a topological manifold has a piecewise linear structure, and when a piecewise linear manifold has a differentiable structure.
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.
In dimensions at most 6 the notions of piecewise linear manifolds and differentiable manifolds coincide.
See also
★ Kirby-Siebenmann class
Reference
★ The wild world of 4-manifolds, , Alexandru, Scorpan, American Mathematical Society, 2005, ISBN 0-8218-3749-4
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