I've been looking at a proof of the Gysin sequence and there the fact above is used, but I do not see how this happens. Probably it comes down to showing that the total space of a disc bundle is homotopyequivalent to the base space, but I do not see how.
2026-04-05 12:08:30.1775390910
Let $\pi : E \to B$ be a disc-bundle. Why does it induce an isomorphism in (co-)homology?
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Not only does $\pi$ induce isomorphisms in (co-)homology, but actually $\pi$ is a homotopy equivalence in the special case that $B$ is a CW complex. To prove this, just use obstruction theory to construct a section $B \to E$, then deformation retract $E$ to the image of that section.
The general case, where $B$ need not be a CW complex, reduces to the case of a CW complex by using the existence of a weak homotopy equivalence $X \mapsto B$ from some CW complex $X$.