Can two small exotic smooth $\mathbb{R}^4$ manifolds be combined as a standard smooth $\mathbb{R}^4$? I just seen De Michelis and Freedman's paper Uncountably many exotic $\mathbf{R}^4$'s in standard 4-space, J. Differential Geometry
35 (1992) pp 219-254, doi:10.4310/jdg/1214447810.
If I understand correctly, they said there are many small exotic $\mathbb{R}^4_\xi$ manifolds which can embedding into a standard $\mathbb{R}^4$.
I can not catch all the ideas in their paper, but I would very like to know the question:
Can two small exotic $\mathbb{R}^4_\xi$ and $\mathbb{R}^4_{\xi'}$ manifolds which embedding in a standard $\mathbb{R}^4$ be combined (annihilated) each other and become a global standard $\mathbb{R}^4$ manifold?
Or can a global standard $\mathbb{R}^4$ manifold be continuously deformed to create two small exotic smooth $\mathbb{R}^4_\xi$ and $\mathbb{R}^4_{\xi'}$ manifolds?
 A: If the "combining" you're referring to is the end-sum, it turns out that no such exotic $\mathbb{R}^4$'s exist.
This was actually shown by Gompf in the appendix to "An infinite set of exotic $\mathbb{R}^4$'s" (Journal of Differential Geometry 1983).
The basic idea is to suppose $R_1 \natural R_2 = \mathbb{R}^4$ and use the Eilenberg swindle to get
$$R_1 = R_1 \natural (\natural_{i=1}^\infty \mathbb{R}^4) = R_1 \natural (\natural_{i=1}^\infty (R_2 \natural R_1)) = R_1 \natural R_2 \natural R_1 \natural R_2 \dots = \mathbb{R}^4 \natural \mathbb{R}^4 \natural \dots = \mathbb{R}^4 $$


[Edit: old note]
I think any such exotic $\mathbb{R}^4$ would have to be standard at infinity.
To see this, suppose two exotic $\mathbb{R}^4$'s $R_1$ and $R_{2}$ have end sum $R_1 \natural R_2$ diffeomorphic to the standard $\mathbb{R}^4$.
The end sum $R_1 \natural R_2$ is constructed by taking smoothly properly embeddings of rays $\gamma_i: [0, \infty) \rightarrow R_i$.
We then take tubular neighborhoods of ray which will be diffeomorphic to $[0,\infty) \times \mathbb{R}^3$.
Delete each of these tubular neighborhoods to get $U_i \subset R_i$.
We then glue $U_1$ and $U_2$ together along the new boundary to get $R_1 \natural R_2$.
When Gompf introduced this in his aforementioned paper, he showed that this produces a well defined smooth manifold homeomorphic to $\mathbb{R}^4$.
If $R_1 \natural R_2$ is diffeomorphic to $\mathbb{R}^4$, then there is a neighborhood of infinity $V \subset R_1 \natural R_2$ that is diffeomorphic to a neighborhood of infinity of $\mathbb{R}^4$, namely $S^3 \times \mathbb{R}$.
This induces a diffeomorphism with a neighborhood of infinity of each $U_i$  with a neighborhood of infinity of $\mathbb{R^3} \times (-\infty,0]$.
We can then glue the neighborhood of the rays $\gamma_i$ back in.
This will induce a diffeomorphism of the neighborhoods of infinity of $R_i$ with a neighborhood of infinity of $\mathbb{R^3} \times (-\infty,0]$ glued with $[0,\infty) \times \mathbb{R}^3$.
This will be the standard $\mathbb{R}^4$ and so $R_i$ is diffeomorphic to $\mathbb{R}^4$ at infinity.
