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.

arbitrarymaps, you must then impose the condition $2\dim M-\dim N +k+1\leq 0$, i.e. $k\leq\dim N-2\dim M -1$, which is exactly the estimate I wrote in my previous comment. By the way, you will find precisely this connectivity estimate on the first page of the introduction of Dax's article. $\endgroup$1more comment