TITLE: Homotopy groups of spaces of embeddings
QUESTION [19 upvotes]: Let $\mathrm{Emb}(M, N)$ be the space of smooth embeddings of a closed manifold  $M$ into a manifold $N$ equipped with smooth compact-open topology. 
 Question 1. Are there conditions ensuring that the $k$th homotopy group of $\mathrm{Emb}(M, N)$ is infinite? Nontrivial? 
I care about the situation when $k>0$ and $1<\dim(M)<\dim(N)$, and most importantly, the homotopy group is based at a homotopy equivalence $M\to N$; let's assume all this is true.
The only  result  I know are in the metastable range (due to Dax who built on Haefliger's work): If $k\le 2\dim(N)-3\dim(M)-3$ and $k\le\dim(N)-\dim(M)-2$, then the inclusion of $\mathrm{Emb}(M, N)$ into $\mathrm{Map}(M,N)$ is $k$-connected. Here $\mathrm{Map}(M,N)$ is space of continuous maps from $M$ to $N$, which is extensively studied in homotopy theory, especially rationally. 
 Question 2. Is anything else known about $\pi_k(\mathrm{Emb}(M, N))$ when $N$ is a the total space of a vector bundle over $M=S^n$, and the homotopy group is based at the zero section? What happens if the vector bundle is trivial, i.e. $N=S^n\times \mathbb R^l$?
I have spent some time reading works of Goodwillie, Klein, and Weiss, see 
 here , who build a framework for analyzing $\mathrm{Emb}(M, N)$. Unfortunately, I was unable to extract any computations that would shed light onto the above questions. It seems the questions are open and hard, is this true? Any references, hints, or heuristics would be greatly appreciated.

REPLY [9 votes]: The details of the result slightly stronger than that oulined in Tom Goodwillie's answer appears as Proposition 3.5 of this paper of Bustamante, Krannich, and myself. Let me state a version of this result that uses a bit less notation that the one in the paper:

Let $M$ be a compact smooth manifold of dimension $d$ and $N \subset \mathrm{int}(M)$ be a compact submanifold. If $N$ has handle dimension at most $d-3$ and the fundamental groups of $M$ are finite at all basepoints, then at all basepoints $\pi_k(\mathrm{Emb}(N,M))$ is finitely generated for $k \geq 2$ and polycyclic-by-finite for $k = 1$.

Note that the condition on the handle dimension of $N$ is in particular satisfied if $N$ has codimension $\geq 3$. This proposition is proven by induction over the embedding calculus Taylor tower.
Furthermore, in even dimension $2n \geq 6$ we also have a result in codimension $\leq 2$, but only for the component of the inclusion under a condition on the complement. This is Theorem C of the paper:

Let $M$ be a compact smooth manifold of dimension $2n \geq 6$ and $N \subset \mathrm{int}(M)$ be a compact submanifold.  If the fundamental groups of $M$ and $M\setminus N$ are finite at all basepoints, then the groups $\pi_k(\mathrm{Emb}(N,M),inc)$ based at the inclusion are finitely generated for $k \geq 2$.

This is deduced using isotopy extension from a statement about diffeomorphism groups. It uses, among other techniques, embedding calculus and the work of Galatius--Randal-Williams and Friedrich on homological stability for diffeomorphisms of high-dimensional manifolds. You can't drop the condition on the fundamental groups, e.g. the component of the inclusion of $\mathrm{Emb}_\partial(D^{d-2},D^d)$ has infinitely generated homotopy groups for $d \geq 6$.