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.