It's been known for a while that primes in number fields can be thought of, from an algebraic point of view, to be similar to knots in 3-manifolds. A good reference (thanks to this question) would be an article by Morishita, 0904.3399.
There are therefore many good analogues of operations, such as covers, or objects, like zeta-functions, that are defined purely algebraically. For example, a linking number of two knots has an easy algebraic definition as the image of one knot in the homology of the complement to the other which is analogous to residue symbol in number theory.
However, the operations of taking connected sum and cutting/gluing along a subsurface don't appear immediately to have an analogue in number fields. If you know how to make sense of "gluing" two schemes and along the "common element , by all means, please tell us!
Either way, here's my question:
What could be an analogue of the Thurston geometrization program for number fields?
(may be this analogue will not be using gluing-like operations after all?)