In the original work of Stiefel and Whitney, characteristic classes were defined as obstructions to the existence of certain fields of linear independent vectors.
The older meaning for obstruction theory in homotopy theory relates to the procedure, inductive with respect to dimension, for extending a continuous mapping defined on a simplicial complex, or CW complex.
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 the image of the 0-skeleton will belong to the same path-connected component of Y.
At some point, say extending the mapping from the (n-1)-skeleton of X to the n-skeleton of X, this procedure might be impossible.
Suppose that B is a simply connected simplicial complex and that p : E → B is a fibration with fiber F. Furthermore, assume that we have a partially defined section σn : Bn → E on the n-skeleton of B.
Because fibrations satisfy the homotopy lifting property, and Δ is contractible; p−1(Δ) is homotopy equivalent to F. So this partially defined section assigns an element of πn(F) to every (n + 1)-simplex.
This cochain is called the obstruction cochain because it being the zero means that all of these elements of πn(F) are trivial, which means that our partially defined section can be extended to the (n + 1)-skeleton by using the homotopy between (the partially defined section on the boundary of each Δ) and the constant map.
Therefore we have a well-defined element of the cohomology group Hn + 1(B; πn(F)) such that if a partially defined section on the (n + 1)-skeleton exists that agrees with the given choice on the (n − 1)-skeleton, then this cohomology class must be trivial.
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 differential structure.
In both cases there are two obstructions for n>9, a primary topological K-theory obstruction to the existence of a vector bundle: if this vanishes there exists a normal map, allowing the definition of the secondary surgery obstruction in algebraic L-theory to performing surgery on the normal map to obtain a homotopy equivalence.