I'm interested in the following collection of questions: Let $S^n\_k = \sqcup\_k S^n$ be a disjoint union of $k$ distinct $n$-dimensional spheres. Write $Emb(S\_k^n, S^{n+2})$ for the space of embeddings of these spheres into $S^{n+2}$. Pick your favorite embedding $e: S\_k^n \to S^{n+2}$, and let $X\_e = S^{n+2} \setminus im(e)$ be the complement of the image of the embedding.


1. What is $\pi\_1(Emb(S\_k^n, S^{n+2}), e)$? Since this is probably unknown, what is known?
2. How is this related to the mapping class group $\pi\_0(Diff(X\_e))$ of $X\_e$?


I ask #2 because in dimension $n=0$, they are the same: the space of embeddings is the configuration space of points in the sphere. Its fundamental group is the (spherical) braid group, which is the same as the mapping class group of the punctured sphere. My guess is that life is not so simple in higher dimensions. Lastly, does any of this simplify out when you get into the range of dimensions where surgery theory starts working well?



Knot complement diffeomorphism groups and embedding spaces
There is a locally-trivial fibre bundle


$$ Diff(S^{n+2}, L) \to Diff(S^{n+2}) \to Emb\_L(\sqcup\_k S^n, S^{n+2})$$


here $Emb\_L(\sqcup\_k S^n, S^{n+2})$ is the component of the link $L$ you're interested in the full embedding space $Emb(\sqcup\_k S^n,S^{n+2})$ and to simplify technicalities, assume $Diff(S^{n+2})$ is the group of orientation-preserving diffeomorphisms of $S^{n+2}$. The bundle is given by restricting a diffeomorphism of $S^{n+2}$ to $L$. $Diff(S^{n+2}, L)$ is the subgroup of $Diff(S^{n+2})$ which preserves the link $L$. 


First observation is that the map $Diff(S^{n+2}) \to Emb\_L(\sqcup\_k S^n,S^{n+2})$ is null-homotopic. It's a simple argument -- isotope your link $L$ to sit in a hemi-sphere of $S^{n+2}$. Then apply a linearization process to linearize (simultaneously) all the diffeomorphisms of $S^{n+2}$ on that hemi-sphere. What I'm claiming is that $Diff(S^{n+2})$ has as a deformation-retract the subgroup that is linear on a fixed hemi-sphere -- so it gives a product decomposition $Diff(S^{n+2}) \simeq SO\_{n+3} \times Diff(D^{n+2})$ (first observed by Morlet, or Cerf, I would guess) among other things. 


So now you have a fibration:


$$\Omega Emb\_L(\sqcup\_k S^n, S^{n+2}) \to Diff(S^{n+2}, L) \to Diff(S^{n+2})$$


where the induced maps on homotopy groups are a short-exact sequence. In particular the fundamental group of your link space injects into $\pi\_0 Diff(S^{n+2}, L)$, and its cokernel is precisely $\pi\_0 Diff(S^{n+2})$. This group is frequently non-trivial as it is the group of exotic $n+3$-sphere provided $n \geq 3$. 


$Diff(S^{n+2}, L)$ is somewhat closely related to $Diff(X\_e)$, especially in high dimensions -- the spherical normal bundle to $L$ is particularly symmetric in low dimensions which causes trouble there. In general the sphereical normal bundle is equivalent to a disjoint union of $S^n \times S^1$, so to make the comparison between $Diff(S^{n+2}, L)$ and $Diff(X\_e)$ you'd need to ask what kind of automorphisms $Diff(X\_e)$ allows on the spherical normal bundle to $L$. There's probably a decent answer to that which doesn't take too much work but the above is a start. 


edit: by Cerf's pseudoisotopy theorem, the kernel of the map $\pi\_0 Diff(S^{n+2}, L) \to \pi\_0 Diff(X\_e)$ contains the exotic sphere "part" of $\pi\_0 Diff(S^{n+2}, L)$. 


