TrickyDicky said:
Yes, not so much as rule out, but certainly it makes one wonder what's going on.
The problem is that "existence" has not yet been well defined.
What topology tells us is that the existence of a global A-field and a global B-field is indeed problematic; that's the reason we have to use R³ \ R instead of R³. A- and B-field are not well-behaved on R³.
This can be seen by using the A-field outside the solenoid
A = \frac{\Phi}{2\pi r^2}(-y,x,0) = \frac{\Phi}{2\pi}\nabla\theta
and taking the limit
r \to 0; \Phi = \text{const.}
Physically this seems to be unproblematic b/c the electrons only "feel" the flux as a physical observable via the phase
e^{i\Phi}
which remains constant. But the A-field as defined above and the gauge function
U = e^{-i\Phi\theta / 2\pi}
from which it can be derived as
A^\prime = 0 \to A = U^\dagger\,(A + i\nabla)\,U
cannot be defined globally; r=0 has to be excluded from our configuration space, i.e. from the base manifold, which results in R³ \ R.
That means that we can define
F = dA = 0
locally but
not globally.
dA = 0
means that A is
locally flat i.e. has vanishing curvature 2-form F (i.e. vanishing el. mag. field strength, i.e. E=0, B=0) locally.
In other words A is pure gauge
locally i.e. A ~ A' = 0 but
not globally.
What Stokes' theorem (and cohomology) tells us is that
\oint_C A
does only depend on the homotopy class of the loop C w.r.t. r=0, i.e. it depends only on the winding number
w[C] = n = 0,\pm 1,\pm 2, ...
of the loop C around the r=0.
Using Stokes' theorem naively
\int_{\partial M} \omega = \int_M d\omega\;\;\to\;\; \int_{C} A = \int_D \nabla\times A = \int_D B = 0
one would derive a vanishing magnetic flux. But this is wrong due to the fact that the extension of A from the loop C to the disc D is not allowed due to the above mentioned singularity in r=0. Therefore the naive application of Stokes theorem is wrong, but this is due to the r.h.s. (= the surface integral) not due to the l.h.s.
Physically one can save the theorem by explaining the B-field via the solenoid, i.e. via replacing the A-field and the B-field by the solenoid solution on a disc D. Mathematically one can save the story by using the locally flat A-field outside and vanishing B-field! Instead one has to cut out the disc D with shrinking radius such that one arrives at R³ \ R where the A-field is well-defined.
That means that the physical solenoid can be replaced by the non-trivial base manifold R³ \ R.
Now the question remains what is physically meaningful. At first glance one could argue in favor of the solenoid with non-vanishing B-field, but even this may be obscure: iff (if and only if) we know that the A-field has been prepared using a solenoid everything is fine. But we (and the electrons) are not able to penetrate the solenoid, so there is no physical mechanism to distinguish between the "physical solenoid" and the "R³ \ R".
What I am saying is that if somebody prepares an impenetrable solenoid with locally flat but non-zero A-field outside w/o telling you any details, you are not able to distinguish between the following two scenarios:
a) the impenetrable solenoid contains a non-vanishing B-field
b) the impenetrable solenoid wraps a topological singularity
