It's well known that with CW hypotheses, the projection of a disk bundle is a homotopy equivalence. But no one seems to make this claim in general, which leads me to believe it might, somehow, be false. Can someone give me a disk bundle $p\colon E \to B$ such that $p$ is not a homotopy equivalence?
I am actually in a somewhat worse situation: I have connected Hausdorff groups $G > H > K$, each a projective limit of Lie groups, one closed in the next, such that $K/H$ is homeomorphic to a Euclidean space, possibly infinite-dimensional, and I am considering the projection $p\colon G/K \to G/H$. I am not sure this is even a bundle, but I have the gut feeling it should be a homotopy equivalence anyway. The theorems of Dold involving numerable covers do not seem to obviously apply, because I am not sure there is anything numerable available. Can anyone say anything about this case?