# Can we decompose Diff(MxN)?

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.

• As a general comment, there's no good reason to expect that the structure of Aut(A x B) is determined by the structure of Aut(A) and Aut(B) in an arbitrary category, where x denotes the product. The product, by definition, only lets you decompose functions into it; it doesn't say anything about how to decompose functions out of it. (This is essentially the same reason that while the maps A -> 1 are trivial by definition, where 1 is the terminal object, the maps 1 -> A can be interesting.) Mar 13, 2010 at 6:57
• (Even in the special case where x is a biproduct, one still has to consider Hom(A, B) and Hom(B, A).) Mar 13, 2010 at 7:18
• From Ryan's comment you can see that just understanding the homotopy type of such a space is not well understood. Mar 29, 2010 at 4:46
• To expand on Qiaochu's comment: in the category of groups, let $A1=B1$ be trivial groups , and let $A2=B2$ be groups of order $2$ . All four of these groups have trivial automorphism groups, but $\operatorname{Aut}(A_1 \times B_1)$ is trivial whereas $\operatorname{Aut}(A_2 \times B_2) \cong \operatorname{GL}_2(\mathbb{F}_2) \cong S_3$. Aug 3, 2010 at 1:50

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.

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.

• Hi John. This result about the fibration being trivial in the stable range after inverting 2, what's the reference for that? Is that Igusa? Jan 24, 2011 at 2:10
• No, It's not Igusa. I do remember Kiyoshi having attributed it to Hatcher. It might be in Hatcher's paper from the 1976 Stanford conference. I realize now that might also have to loop the fibration once to get the correct statement. Jan 24, 2011 at 4:44

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.