This is really hard for me to visualize. What does it look like for a 2-sphere to wrap around the core multiple times? Also, I would have expected it to be able to wrap around in multiple ways since there are more dimensions here, leading to pi^2(b^3 \ {0}) = Z^2. How would one even prove that this isn't the case?
Now, there are many ways of proving that there is only one way (up to homotopy) of wrapping a 2-sphere n times around another 2-sphere, but all of them are fairly involved. The simplest proof comes from an analysis of the Hopf fibration, which roughly describes a relation between the 1-sphere, the 2-sphere and the 3-sphere [1]. Other than this, it follows from the theory of degrees for continuous mappings, or from the Freudenthal suspension theorem and some basic homological computations.
[0] https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres#/me...
You can have “two” spheres wrapped within the Earth with the following parametrization. Using a first coordinate r to denote the distance to the Earth’s center, so that (1,t,z) denotes the points in the sphere of radius 1:
(a,b)-> (1+cos(b)/2, a,b), for a,b in the interval [0,2pi].
Those are not proper spheres (the radius changes) but the surface so parametrized is homotopic to a sphere “counted two times”.
It is not possible to have a warped sphere which does not cross itself, as far as I can tell (but I might be wrong).
The wikipedia image linked by a sibling comment did not help me…
ETA: the issue is not the dimension (2) of your spheres but the codimension (1) inside the object, and the fact that you have only removed the center of the main sphere. I think (caveat emptor) that if you remove 2 points form the solid sphere, you get Z^2. Similar to the case of surfaces and holes.
A Sphere is a Loop of Loops (Visualizing Homotopy Groups)