I am reading Algebraic Topology by E.H.Spanier and in the proof of the Thom-Gysin map for disc bundles (on page 260) he says that $p : E \to B $ is a deformation retraction. I do not understand how this is the case. How do we view $B$ as a subspace of $E$ in the first place ? And then how does $p$ become a deformation retraction ?
Also please advise some reference where I could learn basic properties of disc/sphere bundles. Thanks.
Edit : Here is the statement of the assumption part of Theorem 5.7.11 (in whose proof the statement appears) : Let $(\xi,U_\xi)$ be an oriented q-sphere bundle with base B and projection $\dot{p}=p|_\dot{E}:\dot{E} \to B$. Here $(E,\dot{E})$ is a fiberbundle pair with fiber $(D^{n+1},S^n)$ and $p: E \to B $ is the projection map.
First, I suspect Spanier wants all of his sphere and disc bundles to have linear transition maps, hence there is a canonical zero section $B \hookrightarrow E$. If not, then to make sense of his claim, you need a section; obstruction theory + the fact that $B$ is contractible guarantees that one exists, and indeed you can force it to be in the interior of each fiber.
Once you have a section $s$, note that both $p$ and $s$ are homotopy equivalences by Whitehead + the long exact sequence of homotopy groups of a fibration. Now recall one version of the Whitehead theorem: if $X \hookrightarrow Y$ is a cofibration and also a homotopy equivalence, there is a deformation retraction onto $X$. I would guess that any section of a disc bundle is a cofibration, but if not, pick your section above to be a good one. In any case, once you have this, Whitehead gives you a deformation retraction onto the image of the section, as desired.