Textbook wisdom in electromagnetism tells you that there is no total electric charge on a compact manifold. For example, consider space-time of the form R×M3 where the first factor is time. One defines the total charge via Q(M3)=∫M3⋆j where d⋆F=⋆j is the electric current. If M3 has no boundary (e.g. if it is compact) one can use Stokes' theorem to argue that Q(M3)=∫M3d⋆F=∫∂M3⋆F=0.
I wonder what happens for general four-manifolds M4, especially in the case that the third Betti number is zero (otherwise one can simply integrate over a three-cycle). Is there a sensible way to define charge in the above sense? Can one argue that it has to vanish if ∂M4=0?
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