Suppose that $f:X\rightarrow Y$ is a homotopy equivalence of manifolds. Given a manifold $F$, the pullback construction for $f$ yields a correspondence between isomorphism classes of fibre bundles ...
Problem. How to partition R^3 into pairwise non-parallel lines? A possible solution is to stack infinitely many ``concentric'' hyperboloids, by increasing radius and decreasing slope. And don't ...