Can we decompose Diff(MxN)?

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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.

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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.) – Qiaochu Yuan Mar 13 2010 at 6:57

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(Even in the special case where x is a biproduct, one still has to consider Hom(A, B) and Hom(B, A).) – Qiaochu Yuan 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. – Sean Tilson Mar 29 2010 at 4:46

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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 $\operatorname{Aut}(A_2 \times B_2) \cong \operatorname{GL}_2(\mathbb{F}_2) \cong S_3$. – Pete L. Clark Aug 3 2010 at 1:50

3 Answers

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, here.

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.

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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.

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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?  – Ryan Budney 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. – John Klein 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.

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Your 1st link is broken. – Ryan Budney Mar 13 2010 at 6:52
How odd. I just tried it on 3 different browsers and 2 different computers and it worked each time. I'll try to put it in a comment. web.archive.org/web/20070208084859/http%3A//… – S. Carnahan♦ Mar 13 2010 at 7:21
I get a "403 Forbidden" error message. It appears I'm banned from that webpage. Strange. – Ryan Budney Mar 13 2010 at 7:23

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Perhaps the administrators of archive.org are hockey fans... – S. Carnahan♦ Mar 13 2010 at 7:53

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André has it at his new home: staff.science.uu.nl/~henri105/talbotrefs.html – Ben Wieland May 17 2010 at 21:50
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