Obstruction Theory

Abstract

Recall, in the proof of the Whitehead theorem, we showed that if X is a CW complex and Y is a space with $\pi _{i}(\mathrm{Y}) = 0$ for all i ≥ 0, then any map $\mathrm{f}: \mathrm{X} \rightarrow \mathrm{ Y}$ is homotopic to the base point map $\mathrm{X} \rightarrow \mathrm{ y}_{0} \in \mathrm{ Y}$. The proof was by induction over the skeleta of X. Obstruction theory is a generalization of this technique to the case when the homotopy groups of Y are not necessarily zero. It does not give a complete understanding of when a map $\mathrm{f}: \mathrm{X} \rightarrow \mathrm{ Y}$ is homotopic to a constant, or more generally when two maps $\mathrm{f}_{1},\mathrm{f}_{2}: \mathrm{X} \rightarrow \mathrm{ Y}$ are homotopic, but it gives some insight.