Arithmetic topology of 4-manifolds

We construct a functor from the smooth 4-dimensional manifolds to the hyper-algebraic number fields, i.e. fields with non-commutative multiplication. It is proved that that the simply connected 4-manifolds correspond to the abelian extensions. We recover the Rokhlin and Donaldsons Theorems from the Galois theory of the non-commutative fields.