That’s a good point (though I guess in three dimensions, at least, this issue does not come up since one has a unique smooth structure in this case).

I believe though that if one has smooth oriented structures on both manifolds, one can canonically define a connected sum in this category by arbitrarily choosing a smooth Riemannian metric on both manifolds, and identifying two very small balls in these metrics together using the exponential map (i.e. normal coordinates) for each (and some orientation-preserving orthogonal transformation between the tangent spaces of the centres of the balls), and then removing the interiors of these balls and smoothing things out a bit near the gluing boundary. The configuration space of all the choices one makes here is connected, and the operations are continuous, so I think that this operation is well defined up to smooth diffeomorphism, though I may have missed yet another subtlety here. (I can imagine that if one tries to identify two _large_ balls together, all bets are off.)

]]>two n-spheres; in general the group of components of the diffeomorphisms of a sphere can be quite a large finite group. For example, the Milnor spheres in 7 dimensions can be obtained by gluing together two 7-balls using a non-standard diffeomorphism of their boundaries. I think there is a indeed a canonical way to take the connected sum of

manifolds, but I have a vague recollection that showing it is well defined requires some quite hard theorems such as the annulus conjecture (or rather theorem). ]]>