Obstruction theory

Remarks

The basic facts of obstruction theory are these.

1. The cochain $c(g)$ is a cocycle.
2. The map $g$ extends to $X_{n+1}$ if and only if $c(g) = 0$.
3. The class $[c(g)]$ is zero if and only if the restriction of $g$ to $X_{n-1}$ extends to a map $X_{n+1} \to Y$.

Fact 2 holds because a map $S^n \to Y$ extends over $D^{n+1}$ exactly when it is null-homotopic. Fact 3 is proved with the difference cochain of two maps that agree on $X_{n-1}$.

Example: let $(X, A) = (D^{n+1}, S^n)$, where $X \setminus A$ is a single $(n+1)$-cell, so $X_n = S^n$. Let $n \ge 1$, $Y = S^n$, and let $g$ be the identity of $S^n$. Then $c(g)$ sends the top cell to the generator of $\pi_n(S^n) \cong \Z$, so $g$ does not extend; this is the no-retraction theorem.

Non-example: if $\pi_1(Y)$ acts nontrivially on $\pi_n(Y)$, the value $[g \circ \varphi_e]$ is only defined up to this action, so $c(g)$ is not defined with constant coefficients; one then uses local coefficients.