About the homotopy type of diffeomorphism groups

Question

In this paper by Antonelli, Burghelea and Kahn (Topology, 1972), a homomorphism $L :\pi_i(\operatorname{Diff}(S^n, D_+^{n})) \rightarrow \Gamma^{n+i+1}$ was used as a tool to detect non-triviality of the homomorphism $ \pi_i(\operatorname{Diff}(S^n, D_+^{n})) \rightarrow \pi_i(\operatorname{Diff}(M^n)) $.

Here $D_n^+$ is the upper hemisphere in the $n$-sphere $S^n$. $\operatorname{Diff}(M^n)$ is the group of $C^\infty$ self-diffeomorphisms of $M^n$ with the $C^\infty$ topology; for $X\subset M^n$, $\operatorname{Diff}(M^n,X)$ is its subgroup of diffeomorphisms acting on $X$ as the identity. Finally, $\Gamma^{n+1}$ is the set of diffeomorphism types of smooth manifolds that are homeomorphic to $S^n$ its construction is in page 12-13 of the linked paper.

My questions:

Does there exist a generalization of this result that uses manifolds other than the spheres? In particular, I want to know if there is a homomorphism similar to $L$ that sends $\pi_i(\operatorname{Diff}(M^n))$ to a known space or group,for general $M^n$.

What I'm asking about in the first part of the question is if there are results about detecting non-triviality of some homotopy groups of $\operatorname{Diff}(M^n)$ when $M$ is not a sphere. For the second part of the question, the homomorphism $L$ sends homotopy groups of $\operatorname{Diff}(S^n,D_n^+)$ into the Kervaire-Milnor group $\Gamma^{n+i+1}$. I'm asking if there is a similar homomorphism for general homotopy groups of $\operatorname{Diff}(M^n)$.

Answer 1

$\newcommand{\Diff}{\mathrm{Diff}} \newcommand{\Emb}{\mathrm{Emb}} \newcommand{\Homeo}{\mathrm{Homeo}} \newcommand{\HomEq}{\mathrm{HomEq}}$This is an ancient question, but I suppose it has many answers.

Yes.

There's quite a few tools.

For example, consider the group of diffeomorphisms of the $n$-disc that restrict to the identity on the boundary. Call this $\Diff(D^n)$. There is a 'scanning map'

$$\Diff(D^n) \to \Omega^i \Emb(D^{n-i}, D^n)$$

defined (heuristically) by considering $D^n$ to be a product $D^n = D^i \times D^j$ with $i + j = n$. i.e. we are considering the disc to be fibered by parallel $j$-dimensional discs, in an $i$-dimensional family.

So if you take the induced map on homotopy groups you get a map

$$\pi_k \Diff(D^n) \to \pi_{i+k} \Emb(D^j, D^n).$$

The above map in the $i=1$ case is (what I call) Cerf's scanning homotopy-equivalence.

$$\Diff(D^n) \simeq \Omega \Emb(D^{n-1}, D^n).$$

In general not much is known about these maps when $i>1$. That said, these maps are currently something that are being actively studied.

On the other hand, if you change the manifold a little bit and look at $\Diff(S^1 \times D^n)$ then there are scanning maps of the form

$$\Diff(S^1 \times D^n) \to \Omega^{n-1} \Emb(I, S^1 \times D^n)$$

and these maps can be shown to be non-trivial. That's maybe my current favorite approach, but there are certainly many other maps.

Significantly more elementary, there are the forgetful maps $\Diff(M) \to \Homeo(M) \to \HomEq(M)$, i.e. diffeomorphisms are homeomorphisms are homotopy self-equivalences. It can often be difficult to say things about $\Homeo(M)$, but the space of self-homotopy equivalences of a space can be studied using fairly classical tools.

That particular map studied by Burghelea, Antonelli and Kahn, that looks like it must be an iterated composite of the connecting map in the pseudo-isotopy fibration sequence for $\Diff(D^n)$. That's a fairly specialized tool. That said, the diffeomorphism group of an arbitrary manifold fits into a pseudo-isotopy fiber sequence, and so the connecting map there is perhaps the most directly analogous to your $L$. Specifically, it is the tautological map $\Omega \Diff(M) \to \Diff(I \times M)$, i.e. one is identifying $\Omega \Diff(M)$ with the subgroup of diffeomorphisms of $I \times M$ that preserve the height in the $I$ parameter.