This is about understanding something about what goes on as you go rightward on the table at https://en.m.wikipedia.org/wiki/Homotopy_groups_of_spheres (under General Theory)

There's an applications section in the Wikipedia article, but it's all to other parts of pure math. It's hard to summarize, but they've got to do with obstructions to untangling, unwrapping, or otherwise solving things to do with spaces.