Obstruction class

Definition
Suppose $$X$$ is a topological space with a CW-complex structure, and $$X^k$$ is the $$k$$-skeleton, $$X^{k-1}$$ is the $$(k-1)$$-skeleton. Suppose we have a map from $$X^{k-1}$$ to $$F$$. Then, the obstruction class of this fiber bundle is an element of the cellular cohomology group:

$$H^k(X; \pi_{k-1}(F))$$

defined as follows:

For every $$k$$-cell in $$X$$, we have an attaching map $$S^{k-1} \to X^{k-1}$$. Composing this with the map from $$X^{k-1}$$ to $$F$$, we get a map from $$S^{k-1}$$ to $$F$$, yielding an element of $$\pi_{k-1}(F)$$. Thus, we have a map that associates to every $$k$$-cell an element of $$\pi_{k-1}(F)$$. This is precisely an element of the cellular cochain group $$C^k(X; \pi_{k-1}(F))$$. The cohomology class of this map is the obstruction class.

This cohomology class can be defined in the more general context of a moving target i.e. the problem of finding a section to a fiber bundle over $$X$$ with fiber $$F$$. Here, we require that the fiber bundle is trivial on each of the cells.