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December 23, 2008

Bridge Building

Posted by David Corfield

If anyone wanted to bridge the gap between the two cultures, Terry Tao’s post – Cohomology for dynamical systems might provide a good place to start. Remember our last collective effort at bridge-building saw us rather unsuccessfully try to categorify the Cauchy-Schwarz inequality.

Regarding this current prospective crossing point, we hear that the first cohomology group of a certain dynamical system is useful for the ‘ergodic inverse Gowers conjecture’, and that there are hints that higher cohomology elements may be relevant. The post finishes with mention of non-abelian cohomology.

It wouldn’t be surprising if algebraic topology provided the common ground. A while ago we heard Urs describe Koslov’s work on combinatorial algebraic topology.

Posted at December 23, 2008 1:08 PM UTC

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Re: Bridge Building

Tao’s “dynamical system” (X,(G,))(X,(G,\cdot)) is the action groupoid of GG acting on XX and I think the cohomology groups he describes are the corresponding groupoid cohomology groups.

Yes, this can be interpreted in general nonabelian cohomology.

Nonabelian cohomology, quite generally, is cohomology on \infty-groupoids with coefficients in \infty-groupoids, using homotopical cohomology theory.

One way to model this is using ω\omega-groupoids (sufficient for Tao’s application) and the folk model structure on them.

then an nn-cocycle on a groupoid CC with coefficients in the abelian group AA is an ω\omega-anafunctor from CC to B nA\mathbf{B}^n A, i.e. a span

C^ B nU(1) C \array{ \hat C &\to & \mathbf{B}^n U(1) \\ \downarrow \\ C } where the left leg is an acylic fibration, i.e. an ω\omega-functor which is kk-surjective for all kk.

That ordinary group cohomolgy is reproduced this way is nicely described in the work by Ronnie Brown, Phillip Higgins and Rafael Sivera, in the context of Nonabelian algebraic topology. I know they have a comprehensive monograph in preparation which explains this in detail, one can ask them for a pdf copy. But maybe this is also described in one of their published articles.

Groupoid cocycles such as Tao considers appear in particular in the study of Dijkgraaf-Witten theory in the context of the twisted Drinfeld double. An interpretation of this entirely in the above context of cohomology of \infty-groupoids with coefficients in \infty-groupoids is here.

Posted by: Urs Schreiber on December 23, 2008 4:29 PM | Permalink | Reply to this

Re: Bridge Building

I’m afraid Urs’ comment may scare away people who aren’t experts on ω\omega-groupoids. People usually need to climb the dimensional ladder slowly at first.

So, to build a bridge between cultures, it would be nice to give Terry Tao some good concrete examples where the sort of cohomology class he was asking about actually comes up: an element of H 2(G,X,U)H^2(G,X,U).

I wanted to provide an example, but I didn’t get around to it because I was stunned by his notation H n(G,X,U)H^n(G,X,U). I’d never seen before, so I decided I should get back to my actual work.

Now Urs seems to be bridging the notational gap. So, let me guess: Tao has got a group GG acting on a set XX, and an abelian group UU. So, is H n(G,X,U)H^n(G,X,U) the cohomology of the action groupoid X//GX//G with coefficients in UU?

I guess in his expository post Tao is talking about the case where XX and GG are topologically discrete (indeed finite). So he’s not worrying about the topology of X//GX//G yet. But in many interesting examples of dynamical systems, XX and often GG will have some topology. So, I imagine we’ll shortly need to consider the case of a topological group GG acting on a topological space XX, making X//GX//G into a topological groupoid.

Or, maybe Tao is heading for the world of measure theory rather than topology.

Posted by: John Baez on December 23, 2008 9:59 PM | Permalink | Reply to this

Re: Bridge Building

I’m not sure what you mean by the action groupoid, but if this is the same as the stack X//GX//G, then what Tao describes, although a natural enough object to consider, can’t quite be its cohomology. (However, I should admit that I just gave the post a quick glance.) This is because if G is trivial, then his co-chains are just functions from XX itself, which can’t be enough to give cohomology, even if one puts in the natural condition of being locally constant. When all the fine print is put back in, it should be the cohomology of X//GX//G provided XX is contractible. This case is still interesting since we’re not assuming the action is properly discontinuous. Perhaps this is the situation Tao has in mind.

To get the cohomology of X//GX//G in general, it seems a bit complicated to give the kind of description Tao is using. I guess we need something like a GG-equivariant simplicial covering of XX by contractibles and functions on

G n×G^n \times (various levels of the covering)

Incidentally, Tao considers just UU with trivial action, I think, whereas letting GG act on UU as well is interesting.

Posted by: Minhyong Kim on December 23, 2008 11:17 PM | Permalink | Reply to this

Re: Bridge Building

I was going to define ‘action groupoid’ for you, Minhyong, but then I realized it’s always better to answer questions about definitions by putting the definition in the nnLab and referring to that. We’re getting quite excited about the nnLab around here!

So, I went over to the nnLab and found that ‘action groupoid’ has already been defined.

People who like groupoids like the term ‘action groupoid’; others like ‘weak quotient’ or ‘pseudocolimit’. It’s an incredibly simple thing.

Note that Terry Tao was only talking about the case where XX and GG are finite, which makes the use of ‘stacks’ slightly overkill (unless of course one loves stacks).

Posted by: John Baez on December 24, 2008 2:01 AM | Permalink | Reply to this

Re: Bridge Building

I added a brief remark with links to Lie groupoid differentiable stack to the entry on action groupoid.

Posted by: Urs Schreiber on December 24, 2008 9:59 AM | Permalink | Reply to this

Re: Bridge Building

Hmm, that’s what I thought was meant by X//GX / /G. But then, my question still stands. (I took another look at Tao, and he doesn’t seem to be restricting to finite XX and GG.) If what’s written is the *definition* of the cohomology of X//GX//G, then you seem to get the wrong answer when XX is a complicated space and GG is the trivial group. Of course, I may be mistaken in my belief that X/{e}X/\{e\} is just XX. The way I would have thought of the cohomology of X//GX//G, without getting too stacky, is to consider it as a topological category and take the cohomology of its realization, that is the simplicial space whose 0-simplices are

XX,

1-simplices are

G×XG\times X,

2-simplices are

G× XG×XG\times_X G\times X

and so on. But this doesn’t seem to have the same cohomology as the cochain complex that’s given, *unless* one makes some assumption on XX. I’d have to work it out carefully to say this with confidence, but what Tao has seems roughly to map to the

Posted by: Minhyong Kim on December 24, 2008 11:08 AM | Permalink | Reply to this

Re: Bridge Building

Sorry, I made some transmission error with prrevious post. —————————-

Hmm, that’s what I thought was meant by X//GX / /G. But then, my question still stands. (I took another look at Tao, and he doesn’t seem to be restricting to finite XX and GG.) If what’s written is the *definition* of the cohomology of X//GX//G, then you seem to get the wrong answer when XX is a complicated space and GG is the trivial group. Of course, I may be mistaken in my belief that X/{e}X/\{e\} is just XX. The way I would have thought of the cohomology of X//GX//G, without getting too stacky, is to consider it as a topological category and take the cohomology of its realization, that is the simplicial space whose 0-simplices are

XX,

1-simplices are

G×XG\times X,

2-simplices are

G× XG×XG\times_X G\times X

and so on. But this doesn’t seem to have the same cohomology as the cochain complex that’s given, *unless* one makes some assumption on XX. I’d have to work it out carefully to say this with confidence, but what Tao has seems roughly to map to the E 0E_0 term of a spectral sequence that converges to the correct cohomology, where one takes the filtration conjugate to the usual one. That is, when you have a simplicial space Z={Z n}Z=\{Z_n\}, then there is a spectral sequence whose E 1E_1 is made up of the cohomology of the Z nZ_n’s. What Tao has is related to the E 0E_0 of what’s called the ‘conjugate spectral sequence,’ so that the E 1E_1-term is the cohomology with respect to the `other’ differential that makes up the bi-complex.

Perhaps I’m getting various definitions confused?

Posted by: Minhyong Kim on December 24, 2008 11:11 AM | Permalink | Reply to this

Re: Bridge Building

An important exmple of the conjugate SS
appears in Bott’s work on char classes of foliations

Posted by: jim stasheff on December 25, 2008 1:21 PM | Permalink | Reply to this

Re: Bridge Building

Maybe to further amplify the point about the role played by the topology with examples:

One of Tao’s applications for cocycles on groupoids is to induce from them extensions of the underlying groupoids. He seems to be concentrating on 0- and 1-cocycles in what I have seen, but it might be noteworthy to point out that particularly famous are groupoid 2-cocycles and the central extensions induced by them.

In the case that the groupoid YY in question itself is a hypercover YXY \to X of a topological space XX this is the same as a bundle gerbe on XX. In case that YY is the hypercover of an action groupoid YX//GY \to X//G this is a GG-equivariant bundle gerbe on XX.

These things are well studied by now. now I am wondering how much parallel work there may be hidden under the term “dynamical systems”.

Posted by: Urs Schreiber on December 27, 2008 2:22 PM | Permalink | Reply to this

Re: Bridge Building

(I took another look at Tao, and he doesn’t seem to be restricting to finite XX and GG.)

At the blog entry he says that he is ignoring any topology or measure space structure. So, while XX and GG need not be finite, he seems to be considering the discrete topology on them.

I completely agree that if we do not want to ignore the topology, then one should consider the cohomology in the correct generalized sense. One model would be as the cohomology of the simplicial space obtained as the realization of the nerve of the action groupoid, as you say.

This effectively amounts to considering maps out of hypercovers YX//GY \to X//G in the topological category, the way I tried to indicate.

Posted by: Urs Schreiber on December 27, 2008 11:52 AM | Permalink | Reply to this

Re: Bridge Building

Aha. So XX in his context is acyclic. That makes sense now. It seems then that what he has is also the group cohomology of the abelian group Hom(X,M)Hom(X, M).

Posted by: Minhyong Kim on December 27, 2008 5:47 PM | Permalink | Reply to this

Re: Bridge Building

Yes, that’s the impression I got.

To make further progress, it would be helpful to better understand the aim and purpose of invoking cohomology of “dynamical systems” in the context of what Terry Tao was writing about. I only got a vague impression from what I have seen that they are interested in finding extensions of action groupoids. But I am not sure if that is just for its own sake or if this is to be a means to some other end.

Posted by: Urs Schreiber on December 28, 2008 1:31 PM | Permalink | Reply to this

Re: Bridge Building

So, is H n(G,X,U)H^n(G,X,U) the cohomology of the action groupoid X//GX//G with coefficients in UU?

Yes, that’s what I thought I said in my comment:

Tao’s “dynamical system” (X,(G,))(X,(G,\cdot)) is the action groupoid of GG acting on XX and I think the cohomology groups he describes are the corresponding groupoid cohomology groups.

A familiar example of cocycles on action groupoids arises in Dijkgraaf-Witten theory. Simon Willerton nicely described this in the context of the discrete Freed-Hopkins-Teleman theorem. There it is the action groupoid G//GG//G of the adjoint action of a finite group on itself which matters.

Here this is discussed in the context of general nonabelian cohomology. There are also various figures to be found there which illustrate what is going on geometrically.

we’ll shortly need to consider the case of a topological group GG acting on a topological space XX, making X//GX//G into a topological groupoid.

That case, too, is dealt with in nonabelian cohomology by working with \infty-groupoids internal to the relevant spaces. At differential nonabelian cohomlogy this is described for the smooth case. But the topological or measure-theoretic case works entirely analogously. More on cocycles in this context is at Characteristic classes and forms.

I went over to the nLab and found that ‘action groupoid’ has already been defined.

Yes, and I also gave the link in my reply above.

In fact, in that reply most every technical terms was equipped with a hyperlink to the corresponding nnLab-entry. It’s too bad that nobody noticed that :-(.

I realized it’s always better to answer questions about definitions by putting the definition in the nnLab and referring to that.

Yes. In my last comments and entries, such as the one on Lie II for Lie groupoids, I first put all the relevant information into entries on the nnLab and then sprinkled the relevant links all over the message posted here.

Now I wonder if anyone noticed… But I think it would be great if this is what would eventually become common practice here.

Posted by: Urs Schreiber on December 24, 2008 9:53 AM | Permalink | Reply to this

Re: Bridge Building

good idea
but it would also be good
if we were updated when the n-lab was

Posted by: jim stasheff on December 25, 2008 1:30 PM | Permalink | Reply to this

Re: Bridge Building

If you subscribe to the RSS feed, I believe you will be notified of any changes to nLab (I haven’t tried it yet).

Posted by: Eric on December 25, 2008 5:43 PM | Permalink | Reply to this

Re: Bridge Building

If you subscribe to the RSS feed, I believe you will be notified of any changes to nnLab (I haven’t tried it yet).

Yes. When you subscribe to the nnLab RSS feed by clicking on one of the links provided at nnLab:Feeds your RSS reader will show you all the entries that have been created and/or edited recently.

The list of these recently revided entries can always be found at nnLab:Recently Revised.

Since sometimes it is a bit hard to figure out where the noteworthy changes are hidden among all changes, we keep a commented list of latest changes at nnLab:latest changes.

So the most convenient way to stay in touch with that’s happening at the nnLab might be to subscribe to the RSS feed and then check the feed reader mainly for changes on the page latest changes.

Posted by: Urs Schreiber on December 28, 2008 1:40 PM | Permalink | Reply to this

Re: Bridge Building

The paper – Nonconventional ergodic averages and nilmanifolds – by Kra and Host is mentioned as a source by Tao. Appendix C contains information about the cohomology of dynamical systems.

Perhaps more accessible is his earlier post – What is a gauge?. Example 2 concerns dynamical systems. Surely this material is amenable to Café treatment.

Posted by: David Corfield on December 24, 2008 12:12 PM | Permalink | Reply to this

Re: Bridge Building

The context of this paper seems to make some sense. It is vaguely plausible that if XX is just a measure space, then somehow considerations about the topology of XX go away in the definition of Tao’s complex.

Posted by: Minhyong Kim on December 24, 2008 1:32 PM | Permalink | Reply to this

Re: Bridge Building

Hi, thanks for discussing my post!

While I am ultimately interested in whether higher cohomology has applications to dynamical systems in the measure-theoretic or topological categories, for simplicity I wanted to first start with discrete dynamical systems, i.e. a discrete group G acting on a discrete set X.

My main question is whether higher cohomology has any relationship with the extension problem - i.e. extending a group action of G on X to some principal U-bundle over X, where U is an abelian group. (Nonabelian groups are also very interesting, but for simplicity let’s start with the abelian case. Again, take U to be discrete to avoid complications.)

The first cohomology group $H^1(G,X,U)$ classifies all U-extensions of the G-system X up to conjugation (so I guess it should be equivalent to $Ext(U,G,X)$ or something like that, though one should caution that U is not itself a G-system so one does not precisely have a short exact sequence here). My question was whether the higher cohomology groups, say $H^2(G,X,U)$, are also related to some sort of extension problem. Presumably things like the universal coefficient theorem for cohomology would be relevant here, but other people should be more expert than I in answering this.

Perhaps the situation for group cohomology is better understood - what do higher cohomology groups in that situation have to do with the extension problem for groups?

Posted by: Terence Tao on January 8, 2009 6:49 PM | Permalink | Reply to this

Re: Bridge Building

Here is one short answer: If I understood your formalism correct, your cohomology groups do come with long exact sequences involving the UU. That is, if

0U 1U 2U 300\rightarrow U_1\rightarrow U_2\rightarrow U_3\rightarrow 0

is a short exact sequence of groups, then there should be a long-exact sequence

H 1(G,X,U 2)H 1(G,X,U 3)H 2(G,X,U 1)\rightarrow H^1(G,X,U_2) \rightarrow H^1(G,X,U_3)\rightarrow H^2(G,X,U_1)\rightarrow

Hence, H 2(G,X,U 1)H^2(G,X,U_1) is an obstruction space for lifting a U 3U_3 extension to a U 2U_2 extension. Even though you seem not to be too concerned with them for now, this is especially useful for non-abelian groups, for example, if U 2U_2 is a central extension of U 3U_3. I presume a scenario of this sort will arise when constructing the ‘nilsystems’ you mention.

Posted by: Minhyong Kim on January 8, 2009 9:13 PM | Permalink | Reply to this

Re: Bridge Building

then there should be a long-exact sequence

Just to amplify: there is this long exact sequence in nonabelian cohomology.

The connection homomorphism is constructed, schematically, as follows:

Given the exact sequence

AG^G A\to \hat G \to G

of \infty-groups with AA abelian, form the weak cokernel

coker(AG^) coker(A \to \hat G)

This is weakly equivalent to GG

coker(AG^)G coker(A \to \hat G) \stackrel{\simeq}{\to} G

This means that any GG cocycle gg on an \infty-groupoid XX

X^ g BG X \array{ \hat X &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow^\simeq \\ X }

(with BG\mathbf{B}G the one-object \infty-groupoid corresponding to GG) lifts to a coker(AG^)coker(A \to \hat G)-cocycle g twg_tw just py postcomposition with the weak inverse of the projection.

This g twg_{tw}-cocycle is often called a “twisted cocycle”, even though it technically is still equivalent cohomologycally to gg. For instance the twisted vector bundles appearing in twisted K-theory are such twisted cocycles for the extension of ordinary 1-groups U(1)U(H)PU(H)U(1) \to U(H) \to PU(H).

Now the point to get the connecting homomorphism is that for AA abelian there is a projection

coker(AG^)BA. coker(A \to \hat G) \to \mathbf{B}A \,.

Postcomposing with this projections projects out the “twist” from the twisted cocycle g twg_{tw} to yield a BA\mathbf{B}A-cocycle g obstrg_{obstr}. The class of this is precisely the obstruction to having an (“untwisted”) lift of the original GG-cocycle gg to a G^\hat G-cocycle g^\hat g.

So the connecting homomorphism

H(X,G)H(X,BA) H(X,G) \to H(X, \mathbf{B}A)

is given on cocycles by the composite

gtwistedliftg twprojectionontotwistg obstr. g \stackrel{twisted lift}{\mapsto} g_{tw} \stackrel{projection onto twist}{\mapsto} g_{obstr} \,.

There are some famous examples of this:

- for AG^GA \to \hat G \to G an ordinary central extension of ordinary groups, the twisted lifts are called twisted GG-bundles and the obstructing BA\mathbf{B}A-cocycles are called lifting gerbes

- in particular for the sequence U(1)Spin c(n)SO(n)U(1) \to Spin^c(n) \to SO(n) this yields the Spin-gerbes which obstruct Spin-structures.

- for BU(1)String(n)Spin(n)\mathbf{B}U(1) \to String(n) \to Spin(n) the twisted lifts are twisted String-2-bundles and the obstructiing B 2U(1)\mathbf{B}^2 U(1)-cocycles are the Chern-Simons 2-gerbes obstructing String-structures;

Not so famous yet is the next step:

- for B 5U(1)Fivebrane(n)String(n)\mathbf{B}^5U(1) \to Fivebrane(n) \to String(n) the twisted lifts are twisted Fivebrane-6-bundles and the obstructing B 6U(1)\mathbf{B}^6 U(1)-cocycles are the Chern-Simons 6-gerbes obstructing Fivebrane-structures.

Posted by: Urs Schreiber on January 8, 2009 10:21 PM | Permalink | Reply to this

Re: Bridge Building

Ah, that’s exactly the answer I was looking for, thanks!

Posted by: Terence Tao on January 8, 2009 11:28 PM | Permalink | Reply to this

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