Primary cellular obstruction cochain

Let $n\ge 1$, let $(X,A)$ be a CW pair, and let $f:X^n\cup A\to Y$. Assume either $n\ge 2$, with a fixed extension $\mathcal{P}$ to $X$ of the local system $f^{\ast }{\mathrm{\Pi }}_{n}Y$ on $X^n\cup A$, or $n=1$, with the abelian trivial-conjugation coefficient system specified in the preceding definition.

Supply cellular coefficient coordinates: an orientation and lift of every relative $(n+1)$-cell and the corresponding whisker from its attaching-sphere basepoint to the chosen component coordinate. If ${\mathrm{\Phi }}_e:(D^{n+1},S^n)\to (X^{n+1}\cup A,X^n\cup A)$ is the resulting based characteristic map, define

$\theta (f)(e)=\text{the transport to the chosen cell coordinate of }[f{\mathrm{\Phi }}_e{\mathrm{\mid }}_{S^n}]\in {\pi }_n(Y,f({\mathrm{\Phi }}_e(\ast )))\mathrm{.}$

This assignment is the primary cellular obstruction cochain

$\theta (f)\in C_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^{n+1}(X,A;\mathcal{P})\mathrm{.}$

Reversing the cell orientation negates both its cellular generator and its coordinate value. Changing a lift by a deck transformation changes the generator and the coefficient by the matching monodromy action, exactly as required by the equivariant-Hom rule $\phi (c\cdot g)=g^{-1}\phi (c)$. Thus the cochain is independent of the display coordinates after the canonical basis identification.

By the one-cell extension lemma, $\theta (f)(e)=0$ exactly when $f$ extends over that particular characteristic disk while its map on $X^n\cup A$ is kept fixed. The cocycle and global-choice statements are not part of this definition; they are proved below.