Can we decompose Diff(MxN)?

Question

If you have two manifolds $M^m$ and $N^n$, how does one / can one decompose the diffeomorphisms $\text{Diff}(M\times N)$ in terms of $\text{Diff}(M)$ and $\text{Diff}(N)$? Is there anything we can say about the structure of this group? I have looked in some of my textbooks, but I haven't found any actual discussion of the manifolds $\text{Diff}(M)$ of a manifold $M$ other than to say they are "poorly understood."

Can anyone point me to a source that discusses the manifold $\text{Diff}(M)$? My background is in physics, and understanding the structure of these kinds of groups is important for some of the the things we do, but I haven't seen any discussion of this in my differential geometry textbooks.

Answer 1

The homotopy-type of the group of diffeomorphisms of a manifold are fairly well understood in dimensions $1$, $2$ and $3$. For a sketch of what's known see Hatcher's "Linearization in three-dimensional topology," in: Proc. Int. Congress of. Math., Helsinki, Vol. I (1978), pp. 463-468.

Similarly, the finite subgroups of $Diff(M)$ are well understood in dimensions $3$ and lower. Hatcher's paper is a good reference for that as well, when combined with a few semi-recent theorems.

If you're interested in general subgroups of $Diff(M)$, there's still a fair bit of discussion going on just for subgroups of $Diff(S^1)$, as it contains a pretty rich collection of subgroups.

In high dimensions there's not much known. For example, nobody knows if $Diff(S^4)$ has any more than two path-components. See for example this little blurb. Some of the rational homotopy groups of $Diff(S^n)$ are known for $n$ large enough.

I wrote a survey on what's known about the spaces $Diff(S^n)$, and spaces of smooth embeddings of one sphere in another $Emb(S^j,S^n)$ a few years ago: A family of embedding spaces (Budney, 2006).

Getting back to your earlier question, groups of diffeomorphisms of connect-sums can be pretty compicated objects. In dimension $2$ it's already interesting. For example, $Diff(S^1 \times S^1)$ has the homotopy-type of $S^1 \times S^1 \times GL_2(\mathbb Z)$. Diff of a connect-sum of $g$ copies of $S^1 \times S^1$ has the homotopy-type of a discrete group provided $g>1$, this is called the mapping class group of a surface of genus $g$. It's a pretty complicated and heavily-studied object. In the genus $g=2$ case this group is fairly similar to the braid group on $6$ strands.

In dimension $3$, it's an old theorem of Hatcher's that $Diff(S^1 \times S^2)$ doesn't have the homotopy-type of a finite-dimensional CW-complex, as it has the homotopy-type of $O_2 \times O_3 \times \Omega SO_3$. I've been spending a lot of time recently, studying the homotopy-type of $Diff(M)$ when $M$ is the complement of a knot in $S^3$, and knot complements in general. The paper of mine I linked to goes into some detail on this.

From the perspective of differential geometry, the homotopy-type of $Diff(S^n)$ is rather interesting as it's closely related to the homotopy-type of the space of "round Riemann metrics" on $S^n$. This is a classic construction, is outlined in my paper but it goes like this: $Diff(S^n)$ has the homotopy type of a product $O_{n+1} \times Diff(D^n)$ where the diffeomorphisms of $D^n$ are required to be the identity on the boundary -- this is a local linearization argument. $Diff(D^n)$ has the homotopy-type of the space of round metrics on $S^n$. The idea is that any two round metrics are related by a diffeomorphism of $S^n$. So $Diff(S^n)$ acts transitively on the space of round metrics (with a fixed volume, say), and the stabilizer of a round metric is $O_{n+1}$ basically by the definition of a round metrics. Kind of silly but fundamental.

edit: I should add, there are some nice theorems about $\pi_0 Diff(S^1 \times D^n)$ for $n$ at least 5, and similarly $\pi_0 Diff( (S^1)^n )$, due to Hatcher and Wagoner. They derive their results in some sense indirectly, by getting a strong understanding of the pseudo-isotopy diffeomorphisms of $S^1 \times D^n$. One way to think about their work, is that the isotopy-classes of diffeomorphisms of $S^1 \times D^n$ (fixing the boundary pointwise) is governed by three groups: (1) $\pi_0 Diff(D^n)$, (2) $\pi_0 Diff(D^{n+1})$ and (3) $\pi_0 Emb(D^n, S^1 \times D^n) / Diff(D^n)$. Think of this last group as the isotopy-classes of embedded n-discs in $S^1 \times D^n$ that agree with a standard linear embedding on the boundary. i.e. these are submanifolds without parametrization. It turns out this is a group with a stacking construction. Hatcher and Wagoner show that $\pi_0 Diff(S^1 \times D^n)$ is the direct sum of these three groups, with this group of embedded discs being an infinitely-generated $2$-torsion group.

Recently David Gabai and I were able to give a weak analogue to this Hatcher-Wagoner theorem but in dimension $4$, i.e. for $\pi_0 Diff(S^1 \times D^3)$. While we have not managed any new results about $\pi_0 Diff(D^4)$, we can show $\pi_0 Emb(D^3, S^1 \times D^3)$ is infinitely-generated, even rationally. From a certain perspective our embeddings of $D^3$ in $S^1 \times D^3$ are fairly similar to the Hatcher-Wagoner embeddings, but we rely on slightly different geometry than they do. Roughly speaking, our embeddings stem more from the ability for $S^2 \sqcup S^1$ being able to link in $S^4$, while Hatcher and Wagoner's construction has more to do with $S^i \sqcup S^j$ for $i+j=n$ being able to Hopf link in $S^n$ when $i,j \geq 3$. In a vague sense that is the key difference between our diffeomorphisms being rationally independent, while theirs are $2$-torsion.

Answer 2

When $N$ is the circle there's a sort of answer. In fact there's a whole chapter in the book

Burghelea, Dan; Lashof, Richard; Rothenberg, Melvin: Groups of automorphisms of manifolds. With an appendix ("The topological category'') by E. Pedersen. Lecture Notes in Mathematics, Vol. 473. Springer-Verlag, Berlin-New York, 1975.

dedicated to showing that after looping once, the space $\text{Diff}(M\times S^1)$ splits up to homotopy as (the loops of) $$ \text{Diff}(M\times I) \times B\text{Diff}(M\times I) \times \eta(M) , $$ where the middle term is a non-connective one-fold delooping of $\text{Diff}(M\times I)$ and $\eta(M)$ is the mysterious "nil-term" (when writing $\text{Diff}(W)$ for a manifold $W$ with boundary, the convention is that the diffeomorphisms are to preserve the boundary pointwise). In particular, once gets a decomposition on the level of homotopy groups. (This theorem is an analog of the Bass-Heller-Swan type result which says $K(R[t]) \simeq K(R) \times BK(R) \times \eta(R)$.)

One can say something about the homotopy type the nil-term in the concordance stable range, roughly, $\dim M/3$.

Furthermore, $\text{Diff}(M\times I)$ sits in a fibration sequence $$ \text{Diff}(M\times I) \to C(M) \to \text{Diff}(M) $$ where $C(M)$ is the topological group of concordances of $M$. After inverting 2, this sequence is homotopically trivial and $\pi_k(\text{Diff}(M\times I))$ can be identified with the invariant part of the $\Bbb Z_2$-action on $\pi_k(C(M))$ induced by conjugating a concordance with the diffeomorphism which turns $M\times I$ upside down ($(x,t) \mapsto (x,1-t)$). Lastly, $\pi_k(C(M))$ can be studied via algebraic $K$-theory methods when $k$ is within the concordance stable range.

Answer 3

I'm not an expert, but my impression was that you can't reasonably expect anything like a decomposition in general. Here is a big list of references on automorphisms of manifolds, compiled by Andre Henriques. The wikipedia page has a brief discussion of diffeomorphism groups, and it requires somewhat less background.