Homotopy groups of spaces of embeddings

Question

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.

Answer 1

Suppose that $M$ is compact, $N$ is simply connected and has finitely generated homology, and the codimension $n-m$ is at least $3$. Then the space $Emb(M,N)$ is such that

(1) for every basepoint $\pi_1$ is solvable and $\pi_k$ is finitely generated for all $k\ge 1$.

This can be proved using the Weiss tower and the "analyticity" or "multiple disjunction" result of John Klein and myself. Or it is possible to give an argument that does not use the tower. Either way, you repeatedly use the fact that when (1) holds for the base of a fibration and for every fiber then it also holds for the total space, and the fact that if (1) holds for the base and the total space of a fibration then it holds for every fiber.

Answer 2

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 $d \geq 6$.

Answer 3

Here are some comments:

1) Concerning finiteness results for spaces of embeddings, here is what I remember. The layers of the Goodwillie-Weiss tower when $M^m$ is closed and $N= \Bbb R^n$ have finitely generated homotopy groups when $2m+2 \le n$ (roughly the Whitney range). Furthermore, these fibers are simply connected. It follows that the embedding space $E(M,\Bbb R^n)$ is of finite type (meaning that it is weak equivalent to a CW complex with finitely many cells in each degree). This will imply finite generation of the homotopy groups.

The reason the layers have finitely generated homotopy groups is that they are section spaces over a finite complex where the fibers are built out of configuration spaces by a homotopy inverse limit procedure.

A more basic related result is this: Let $F(X,Y)$ be the function space of maps $X\to Y$, where $X$ is a finite complex and $Y$ is 1-connected with finitely generated homotopy groups. Then $F(X,Y)$ has finitely generated homotopy groups in each degree. One can see this by induction on the cells of $X$.

I think that the same result above holds when $N$ is compact, possibly with boundary and $1$-connected.

I conjectured that the each component of the embedding space should be of finite type even without the hypothesis $2m+2\le n$ (but with $m \le n-3$). However, I do not know how to prove this more general statement.

2) Concerning your first question: if $M \to N$ is a homotopy equivalence, it seems to me that the homotopy fiber $E^\text{pd}(M,N)\to F(M,N)$ taken at your given homotopy equivalence is contractible. Here, $E^\text{pd}(M,N)$ is the space of Poincare embeddings.

Next, one can analyse the difference between the Poincare embedding and the block embedding spaces as a space of lifts of the Spivak bundle. More precisely, the difference is given by factorizations of $$ M \to BG(n{-}m) \times_{BG} BO $$ through $BO(n{-}m)$. Here $BG$ classifies stable spherical fibrations, $BG(k)$ classifies $(k-1)$-spherical fibrations, and the displayed target is meant to be a homotopy pullback. Rationally, the space of such lifts can probably be computed in terms of the cohomology of $M$ (note: $BG$ is rationally trivial).

Finally, the difference between block embeddings and smooth embeddings is given by concordance embedding spaces. In the concordance stable range, one usually analyses these via relative algebraic $K$-theory (a la Waldhausen).

3) A very different approach to computations of $\pi_0(E(M,N))$ appears in my paper On embeddings in the sphere, Proc. Amer. Math. Soc. 133 (2005), 2783-2793.