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. It is traditionally called Eilenberg obstruction theory, after Samuel Eilenberg. 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, 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. Extending from the 1-skeleton to the 2-skeleton means defining the mapping on each solid triangle from X, given the mapping already defined on its boundary edges. Likewise, then extending the mapping to the 3-skeleton involves extending the mapping to each solid 3-simplex of X, given the mapping already defined on its boundary. At some point, say extending the mapping from the -skeleton of X to the n-skeleton of X, this procedure might be impossible. In that case, one can assign to each n-simplex the homotopy class of the mapping already defined on its boundary,. These assignments define an n-cochain with coefficients in. Amazingly, this cochain turns out to be a cocycle and so defines a cohomology class in the nth cohomology group of X with coefficients in. When this cohomology class is equal to 0, it turns out that the mapping may be modified within its homotopy class on the -skeleton of X so that the mapping may be extended to the n-skeleton of X. If the class is not equal to zero, it is called the obstruction to extending the mapping over the n-skeleton, given its homotopy class on the -skeleton.
Suppose that is a simply connected simplicial complex and that is a fibration with fiber. Furthermore, assume that we have a partially defined section on the -skeleton of. For every -simplex in, can be restricted to its boundary. Because send each of these back to each, we have a map from an -sphere to. Because fibrations satisfy the homotopy lifting property, and is contractible; is homotopy equivalent to. So this partially defined section assigns an element of homotopy group| to every -simplex. This is precisely the data of a -valued simplicial cochain of degree on, i.e. an element of. This cochain is called the obstruction cochain because it being the zero means that all of these elements of are trivial, which means that our partially defined section can be extended to the -skeleton by using the homotopy between and the constant map. The fact that this cochain came from a partially defined section can be used to prove that this cochain is a cocycle. If one started with a different partially defined section that agreed with the original on the -skeleton, then one can also prove that the resulting cocycle would differ from the first by a coboundary. Therefore we have a well-defined element of the cohomology group such that if a partially defined section on the -skeleton exists that agrees with the given choice on the -skeleton, then this cohomology class must be trivial. The converse is also true if one allows such things as homotopy sections, i.e. a map such that is homotopic to the identity map on. Thus it provides a complete invariant of the existence of sections up to homotopy on the -skeleton.
Applications
By inducting over, one can construct a first obstruction to a section as the first of the above cohomology classes that is non-zero.
This can be used to find obstructions to trivializations of principal bundles.
Because any map can be turned into a fibration, this construction can be used to see if there are obstructions to the existence of a lift of a map into to a map into even if is not a fibration.
It is crucial to the construction of Postnikov systems.
