Let $W$ be $B^4$ or $S^3 \times I$. Let $Y$ be a properly embedded surface in $W$. Let $f : W \to W$ be a diffeomorphism which is the identity near $\partial W$. Very little is known about $\pi\_0(\mbox{Diff($W$) rel $\partial W$})$ (see [page 4 of this survey by Hatcher](http://pi.math.cornell.edu/~hatcher/Papers/Diff(M)2012.pdf)), so $f$ might not be isotopic to the identity (or, in the $S^3\times I$ case, isotopic to a standard twist diffeomorphism). Nevertheless, I believe it is true that $f(Y)$ is isotopic rel boundary to $Y$. In other words, exotic diffeomorphisms cannot do anything exotic to embedded surfaces.


The proof I have in mind (based on conversations with an expert) uses a "push-pull" type argument (and, in the $S^3\times I$ case, straightening along an arc) to concentrate any potential exoticness of $f$ in the neighborhood of one or two points. We can take these points to be far from $Y$, and it follows that $f(Y)$ is isotopic to $Y$.


Rather than write down the details of this proof, I would prefer to just cite a published result. So my question is:



> 
> Is there a citable reference for the above claim that $f(Y)$ is
>  isotopic rel boundary to $Y$? Is there a citable reference for the claim that diffeomorphisms of simple 4-manifolds (like $B^4$ or $S^3 \times I$) can be "tamed" in the complement of a finite number of points?
> 
> 
> 



Can an exotic diffeomorphism of the 4-ball change the isotopy class of an embedded surface?
Isn't this relatively obvious for $W=B^4$ (i.e. it only took me several hours to realize it was trivial)? Isotope $Y$ to $Y'$ by an isotopy $g\_t$ into a small collar neighborhood of $\partial W$ (which we can do by general position), then $f(Y')=Y'$ since you've assumed $f$ is the identity in a neighborhood of $\partial W$. But also $f(Y)$ is isotopic to $f(Y')=Y'$ by the conjugate isotopy $fg\_tf^{-1}$, hence $f(Y)$ is isotopic to $Y$. 


For $S^3\times I$ one can probably also achieve this by the lightbulb trick, but I haven't thought it through carefully. 


