# Obstruction class

## Definition

Suppose is a topological space with a CW-complex structure, and is the -skeleton, is the -skeleton. Suppose we have a map from to . Then, the **obstruction class** of this fiber bundle is an element of the cellular cohomology group:

defined as follows:

For every -cell in , we have an attaching map . Composing this with the map from to , we get a map from to , yielding an element of . Thus, we have a map that associates to every -cell an element of . This is precisely an element of the cellular cochain group . 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 with fiber . Here, we require that the fiber bundle is trivial on each of the cells.