I'm looking at Hatcher's chapter on spectral sequences and can't tease out the meaning of a statement early in the proof of the existence of the Serre spectral sequence (on homology). The goal at this step in the proof is to produce the diagram:
$$\begin{matrix} \bigoplus\limits_{\alpha}H_{p+q}(\widetilde{D}_{\alpha}^p,\widetilde{S}_{\alpha}^{p-1};G) & \xrightarrow{\widetilde{\Phi}_*} & H_{p+q}(X_p,X_{p-1};G) \\
\oplus_{\alpha} \epsilon^p_{\alpha} \downarrow \;\cong & & \Psi \downarrow \; \cong \\
\bigoplus_{\alpha} H_q(F;G) & \cong & H_p(B^p,B^{p-1};\mathbb{Z}) \otimes H_q(F;G)
\end{matrix}$$
where we do so by first looking at the characteristic map for the "$\alpha^{th}$" $p$-cell, $e^p_{\alpha}$ in $B^p$, given below $$\Phi_{\alpha}: D_{\alpha}^p \rightarrow B^p$$
After this, Hatcher says some mysterious stuff about the restriction of $\Phi_{\alpha}$ to the boundary sphere and interior of $D_{\alpha}^p$ before defining $\widetilde{D}_{\alpha}^p:=\Phi_{\alpha}^*(X_p)$, which he calls the "pullback fibration over $D_{\alpha}^p$."
I'm having a hard time finding out what $\widetilde{D}_{\alpha}^p$ is. My understanding of the pullback is that it only makes sense when you have two (or more, often a homotopy's worth of) maps. I think it's possible that the restriction maps (with inclusion) could come into play here, but as I understand it, you're not going to get anything in your pullback if a pair of maps in your collection have disjoint codomains, which they would be for the restriction maps $\Phi_{\alpha}|_{\partial D_{\alpha}^p}$ and $\Phi_{\alpha}|_{D_{\alpha}^p-\partial D_{\alpha}^p}$. It might make sense to talk about the pullback across all the characteristic maps, but that doesn't seem to be what he's saying. Does anyone know what this "pullback fibration" might be?
Best Answer
If $f\colon X \to Z$ and $g\colon Y \to Z$ are any continuous maps then the pullback of $Y$ by $f$ is defined by $$f^*(Y) = X\times_{Z} Y = \{ (x, y) \in X\times Y \mid f(x) = g(y)\}$$ If $g$ is a fibration then the projection map $f^*(Y) \to X$ is also a fibration, where the fibre over $x \in X$ is canonically homeomorphic to the fibre over $f(x)$.
In this case Hatcher is taking the fibration $\pi\colon X \to B$ and first restricting it to the $p$-skeleton $B^p$ to get the fibration $\pi_p\colon X_p \to B^p$ (this is technically the pullback along the inclusion map $B^p \to B$), and then pulling that fibration back to the disk $D_\alpha ^p$ via its characteristic map $\Phi_\alpha \colon D_\alpha^p \to B^p$. In symbols,
$$ \tilde{D}_\alpha^p = \Phi_\alpha^*(X_p) $$
which is isomorphic to just doing $\Phi_\alpha^*(X)$.