In a fiber bundle, when we interchange the base space and fiber, does it result in a different total space? What if the base space and fiber are both spheres?
Take the bundle $S^1 \hookrightarrow S^3 \rightarrow S^2$. Here $S^1$ is the fiber and $S^2$ is the base space. If I now consider bundle $ S^2 \hookrightarrow X \rightarrow S^1$, my question is whether $X$ is in anyway related to $S^3$, which is the total space before the interchange of the fiber and bases space. Of course, there is a trivial bundle $ S^2 \hookrightarrow X \rightarrow S^1$ but I am asking about any non-trivila ones.