This is a variation on: https://mathoverflow.net/questions/21742/knot-complement-diffeomorphism-groups-and-embedding-spaces] question for another type of very simple manifold.

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?

This is a variation of Craig's 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?