Obstruction class

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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.