TITLE: Are "Unions" of small exotic $\mathbb{R}^4$'s small?
QUESTION [5 upvotes]: Suppose $M$ is a smooth 4-manifold, and $U,V \subset M$ are exotic $\mathbb{R}^4$'s, i.e. homeomorphic to standard $\mathbb{R} ^4$.
Further more suppose $U$ an $V$ intersect nicely sucht that $U \cup V \subset M$ is again homeomorphic to $\mathbb{R}^4$.
An exotic $\mathbb{R}^4$ is called small if it embeds into $S^4$.
Now my question:
If $U$ and $V$ are both small, is $U \cup V $ then also small?
I thought about that for quite a long time, somehow I believe it should be true, but do not see any way to prove it.

REPLY [3 votes]: The answer is yes.  You can embed U and V disjointly in $S^4$, and then do the connected sum at infinity, or `end-connected sum', using an arc going from one to the other. This construction is, I guess, what you intend by $U \cup V$, and is discussed for example in Gompf's paper, Three exotic R4's and other anomalies, J. Differential Geom. 18 (1983), no. 2, 317–328.