Obstruction theory
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In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.

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
• Surgery theory, where one considers the "surgery obstruction"

References

• Scorpan, Alexandru (2005). The wild world of 4-manifolds. American Mathematical Society. ISBN 0-8218-3749-4. 

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