This is a variation of Craig's [Knot complement diffeomorphism groups and embedding spaces](https://mathoverflow.net/questions/21742/knot-complement-diffeomorphism-groups-and-embedding-spaces]) for a different type of very simple manifold (surfaces which have a 1-relator fundamental group instead of high dimensional spheres).


Let $X$ be a smooth manifold of dimension $2n \geq 6$. Write 
$$Emb(\Sigma\_g,X)$$ 
for the space of smooth embeddings of genus $g$ surface into $X$. Pick any embedding $\phi:\Sigma\_g \to X$, and let $X\_\phi=X \backslash \phi(\Sigma\_g)$ be the complement of the image of the embedding.


**Question.** What can be said about $\pi\_k(Emb(\Sigma\_g,X))$ and its relation to the homotopy groups $\pi\_{k-1} \mathrm{Diff}(X\_\phi)$ from the point of view of surgery, disjunction etc? 



Embedding spaces and surface knots in high dimensional manifolds
Here are some general comments. We can let $\Sigma$ be any closed smooth $k$-manifold and let $X$ be any smooth $n$-manifold. 


Fix a basepoint embedding $\Sigma \to X$.
Let $N$ be a compact regular neighborhood of $\Sigma$ in $X$. Then the procedure of assigning the normal bundle to an embedding defines a homotopy fiber sequence
$$
E(N,X) \to E(\Sigma,X) \to \text F(\Sigma, BO(n-k))
$$
($E$ = smooth embeddings, $F =$ functions) where $E(N,X)$ is the fiber over the point of $F(\Sigma, BO(n-k))$ represented by the normal bundle of $\Sigma$ in $X$. From this point-of-view we regard the difference between $E(N,X)$ and $E(\Sigma,X)$ as understood and we choose to work instead with $E(N,X)$.


Let $C$ be the closure of the complement of $N$ in $X$. Then there is a fiber sequence
$$
\text{Diff}(C) \to \text{Diff}(X) \to E(N,X)
$$
where $\text{Diff}(C)$ denotes the diffeomorphisms of $C$ which preserve the boundary point-wise. 


On homotopy groups the last displayed fiber sequence gives a long exact sequence 
$$
\cdots\to \pi\_j\text{Diff}(C) \to \pi\_j\text{Diff}(X) \to \pi\_jE(N,X) \to \pi\_{j-1}\text{Diff}(C) \to \cdots
$$ 


