For each finite dimension n (1, or 2, or 3, etc…), the sphere in dimension n can’t be contracted because of that empty n-dimensional space it surrounds. But that same sphere is the “equator” of the sphere in the next higher dimension, n+1. There, the n-dimensional equator can contract along one of the hemispheres, to a pole. But then that whole (n+1)-dimensional sphere still isn’t contractible, because of the (n+1)-dimensional space it surrounds.
BUT the (n+1)-dimensional sphere can contract along one of the hemispheres in the (n+2)-dimensional sphere. And so on.
For any particular finite dimension n, there is an n-dimensional obstruction to contracting the sphere in that dimension. But if you go all the way to infinitely-many dimensions, there is no obstruction that ever stops contractibility of the infinite-dimensional sphere.
For each finite dimension n (1, or 2, or 3, etc…), the sphere in dimension n can’t be contracted because of that empty n-dimensional space it surrounds. But that same sphere is the “equator” of the sphere in the next higher dimension, n+1. There, the n-dimensional equator can contract along one of the hemispheres, to a pole. But then that whole (n+1)-dimensional sphere still isn’t contractible, because of the (n+1)-dimensional space it surrounds.
BUT the (n+1)-dimensional sphere can contract along one of the hemispheres in the (n+2)-dimensional sphere. And so on.
For any particular finite dimension n, there is an n-dimensional obstruction to contracting the sphere in that dimension. But if you go all the way to infinitely-many dimensions, there is no obstruction that ever stops contractibility of the infinite-dimensional sphere.