Obstruction theory

Summary

Obstruction theory studies when a map defined on part of a CW complex extends to the whole complex, or when a fibration has a section. It works by extending one skeleton at a time. At each step the problem of going one cell dimension higher is measured by a cohomology class, the obstruction. The method turns geometric extension problems into computations in singular cohomology with coefficients in homotopy groups. Characteristic classes such as the Euler class and the Stiefel–Whitney classes arise as obstructions of this kind.