Embedding spaces and surface knots in high dimensional manifolds

Question

This is a variation of Craig's 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?

Answer 1

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 $$ \cdots\to \pi_j\text{Diff}(C) \to \pi_j\text{Diff}(X) \to \pi_jE(N,X) \to \pi_{j-1}\text{Diff}(C) \to \cdots $$