(The question will be interesting for topological/Pl as well but in order to not be too vague I will restrict the meaning of manifold to smooth manifold without boundary).
I'm interested in the function $f: \mathbb{N} \times\mathbb{Z}_{\ge -1} \to \mathbb{N}$ defined as follows:
$$f(n,k) := \inf_{r \in \mathbb{N}}\{\pi_i(Emb(M,\mathbb{R}^r) = 0, \forall i\le k, \forall \text{ $M$ compact connected n-manifold} \}$$
In other words, $f(n,k)$ is the smallest integer $r$ satisfying that for any compact connected $n$-dim manifold $M$ the space $Emb(M,\mathbb{R}^r)$ is $k$-connected.
Examples:
$f(n,-1)$ - This is the smallest integer s.t. any compact connected $n$-manifold can be embedded in $\mathbb{R}^{f(n,-1)}$. The whiteny embedding theorem tells us that $f(n,-1) \le 2n$. This is also the best linear bound since suitable real projective spaces provide counter examples.
$f(n,0)$ - This is the smallest integer s.t. any compact n-manifold has a unique isotopy class of embeddings in $\mathbb{R}^{f(n,0)}$. I think its known (though I don't have a reference) that if $n \ge 2$ then $f(n,0) \le 2n+1$ and I think this is the best possible linear bound here as well (please correct me if this is wrong).
I know this is not a lot of data to go on but it seems reasonable to conjecture that $f(n,k)$ has a linear bound in both $n$ and $k$. Hence:
Question: Is $f(n,k) = O(n,k)$? If so what is the smallest linear bound on $f(n,k)$ in both variables? Is it perhaps $2n+k+1$? (insert optimistic smile).
