Group Cocycles and Simplices

September 22, 2008

Posted by Urs Schreiber

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Christoph Wockel asks me to forward the following question to the $n$ -Café:

What is a reference for a generalisation of the following canonical cocycle to higher dimensions?

Let $G$ be a connected (locally contractible) topological group and take a section $a : G \to P G,$ continuous on a neighbourhood of the neutral element, of the endpoint evaluation map $ev : P G \to G$ from the space $P G$ of continuous paths starting at the neutral element of $G$ . Then the assignment $(g, h) \mapsto [a (g) + g \cdot a (h) - a (g h)]$ is a $π_{1} (G)$ -valued group 2-cocycle on $G$ , describing the universal cover of $G$ as a central extension. Moreover, this cocycle is universal for discrete groups.

A similar construction works in higher dimensions, yielding for each $(n - 1)$ -connected group $G$ a $π_{n} (G)$ valued group $(n + 1)$ -cocycle. Moreover, this cocycle is universal for discrete groups. I’ve been searching for a reference for this, but did not succeed.

Thanks for any hints.

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My remark: notice that the construction this question is about is closely related to the construction of Čech cocycles for characteristic classes by Brylinski and McLaughlin which I talked about here.

Posted at September 22, 2008 9:50 AM UTC

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Re: Group Cocycles and Simplices

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You’re saying there’s a god-given $π_{n} (G)$ -valued $(n + 1)$ -cocycle on a sufficiently nice topological group $G$ .

Instead of focusing on the specific cocycle, let’s think about its cohomology class. Then you’re saying that if $G$ is sufficiently nice, there’s a god-given element of

$H^{n + 1} (G, π_{n} (G))$

This is group cohomology. But this is isomorphic to

$H^{n + 1} (K (G,1), π_{n} (G))$

where $K (G,1)$ is the 1st Eilenberg–Mac Lane space of $G$ .

But cohomology is represented by Eilenberg–Mac Lane spaces, so the above group is isomorphic to

$[K (G,1), K (π_{n} (G), n + 1)]$

where the square brackets mean ‘the set of homotopy classes of maps’.

I would like to keep chewing away on this until I see that the god-given element you’re talking about is something like an identity map in disguise! But I seem to be stuck right now.

If we take advantage of your assumption that the topological group $G$ is $(n - 1)$ -connected, then we can use the Hurewicz theorem, which says that in this case

$π_{n} (G) ≅ H_{n} (G)$

I’m not sure how this helps, but I thought I should mention it.

Posted by: John Baez on September 23, 2008 3:55 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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If $G$ is a topological group, what is a $K (G,1)$ ? Did you mean $BG$ ?

There are spaces where the fundamental group isn’t discrete (Hawaiian earring, say), but that’s getting a bit much :)

Posted by: David Roberts on September 24, 2008 12:57 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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David wrote:

If $G$ is a topological group, what is a $K (G,1)$ ? Did you mean $B G$ ?

No, I meant the classifying space of the underlying group $G_{disc}$ of the topological group $G$ :

$K (G,1) : = B G_{disc}$

For lurking nonexperts, let me expand on this.

For any sufficiently nice topological group $G$ , there is a classifying space $B G$ such that isomorphism classes of principal $G$ -bundles over a paracompact space $X$ are in 1-1 correspondence with homotopy classes of maps

$X \to B G$

When we’ve got a discrete topological group $G$ — or in other words, just a plain old group — we usually call the classifying space of $G$ an Eilenberg–Mac Lane space $K (G,1)$ . This may alternatively be described as the pointed connected space with $π_{1} = G$ and all higher $π_{n}$ ’s trivial.

Any topological group $G$ has an underlying discrete group, say $G_{disc}$ . There’s a continuous homomorphism

$G_{disc} \to G$

and by functorial abstract nonsense this gives a map of pointed spaces

$B G_{disc} \to B G$

If we agree that the only sensible meaning of $K (G,1)$ is $B G_{disc}$ , then we can say this is a map

$K (G,1) \to B G$

When $G$ is a Lie group, here’s a nice way to think about this. $B G$ is the classifying space for $G$ -bundles, while ${BG}_{disc} = K (G,1)$ is the classifying space for $G$ -bundles equipped with a flat connection — or flat $G$ -bundles, for short. The map

$K (G,1) \to B G$

comes from the fact that any flat $G$ -bundle gives a $G$ -bundle.

There are very nice relationships between this stuff and ‘secondary characteristic classes’, like Chern–Simons classes.

I wish I understood $B G_{disc}$ better even for quite simple Lie groups, like $G = U (1)$ and $G = SU (2)$ .

For example: what’s the cohomology of $B G_{disc}$ in these cases? I know some nice elements: secondary characteristic classes. But what’s the whole story?

Posted by: John Baez on September 24, 2008 7:21 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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When $G$ is a Lie group, here’s a nice way to think about this. $B G$ is the classifying space for $G$ -bundles, while $B G_{disc} = K (G,1)$ is the classifying space for G-bundles equipped with a flat connection — or flat $G$ -bundles, for short.

Incidentally, this is the answer to the question I asked here on January 29 and, after Jim made a remark, again on January 30. Thanks for the reply!! :-)

This was the context of my question back then: there are nice models for the rational approximation of $K (G,1)$ for $G$ any $∞$ -Lie group which are

- (generalized) smooth.

And this is closely related to $∞$ -Lie integration and various other things:

namely, given an $L_{∞}$ -algebra $g$ , there is a generalized smooth space (a sheaf on manifolds) which I keep calling by the weird name $S (CE (g))$ which is the smooth classifying space for flat $g$ -valued diffrential form data – or in other words: trivial $G$ - $∞$ -bundles with chosen smooth flat $∞$ -connection.

In particular, the fundamental smooth $∞$ -groupoid of these spaces is the one-object $∞$ -groupoid coming from the simply connected smooth $∞$ -group $G$ integrating $g$ .

$G = {Aut}_{•} (Π_{∞} (S (CE (g)))) .$

This maybe slightly scary chain of symbols is nothing but the analog of

$G = π_{1} (K (G,1))$

for $G$ an ordinary group.

Posted by: Urs Schreiber on September 25, 2008 12:07 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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You’re saying there’s a god-given $π_{n} (G)$ -valued $(n + 1)$ -cocycle on a sufficiently nice topological group G.

Instead of focusing on the specific cocycle, let’s think about its cohomology class. Then you’re saying that if G is sufficiently nice, there’s a god-given element of

…

(1)

[K (G,1), K (π_{n} (G), n + 1)]

where the square brackets mean ‘the set of homotopy classes of maps’.

Here $G$ is $(n - 1)$ -connected, so $K (G,1) = BG$ is $n$ -connected with $π_{n + 1} (BG) = π_{n} (G)$ . So the first non-trivial stage in the Postnikov tower is

(2)

K (G,1) \to K (π_{n} (G), n + 1),

exactly the sort of map you are looking for.

Posted by: Dan Christensen on September 24, 2008 3:03 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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I wrote:

Here $G$ is $(n - 1)$ -connected, so $K (G,1) = BG$ is $n$ -connected …

I was jumping to the conclusion that when John wrote $K (G,1)$ for a topological group $G$ , he meant $BG$ , but from a later message of John’s I see that he actually meant to regard $G$ as a discrete group when it appears in the expression $K (G,1)$ .

My message gave a map $BG \to K (π_{n} (G), n + 1)$ , and John points out that there is a natural map $K (G,1) \to BG$ , so composing these must give the natural map

(1)

K (G,1) \to K (π_{n} (G), n + 1)

that is being looked for.

Posted by: Dan Christensen on September 26, 2008 1:15 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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I should add that I have the vague feeling you’re describing the Postnikov tower of $G$ , or something like that.

The Postnikov tower of a connected space $X$ describes this space in terms of the groups $π_{n} (X)$ and certain ‘Postnikov invariants’

$k_{n} \in H^{n + 1} (X_{n - 1}, π_{n} (X))$

where $X_{n - 1}$ is $X$ with its homotopy groups above the $(n - 1)$ st killed.

The funny pattern of $n$ , $n + 1$ and $n + 2$ in the above formula reminds me of your sentence

… yielding for each $(n - 1)$ -connected group $G$ a $π_{n} (G)$ valued group $(n + 1)$ -cocycle.

If you’ve never thought about Postnikov towers, and you like higher categories, there might conceivably be someplace to start learning about them that’s even worse than my lectures on cohomology and $n$ -categories.

Posted by: John Baez on September 23, 2008 7:08 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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I should add that I have the vague feeling you’re describing the Postnikov tower of $G$ , or something like that.

I still haven’t quite found the time to think about this in the detail that I ought to, but it seems that this is a way of looking at the construction of higher String-like extensions by successively killing homtopy groups.

In fact, if I think about it for a bit it feels that I should be able to translate this directly to the construction of the strict String 2-group and of the strict Fivebrane 6-group by $L_{∞}$ -integration in those notes on Twisted nonabelian differential cohomology that some of you have seen.

One nice aspect of Christoph’s point of view, in contrast to the $L_{∞}$ -integration picture, is that it allows to talk not only about killing of non-torsion homotopy groups, but also about the torsion homotopy groups.

In particular, while I can express the steps $Spin (n) \mapsto String (n) \mapsto Fivebrane (n)$ in terms of $∞$ -Lie theory, I can’t look at it this way for the first two steps $O (n) \mapsto S O (n) \mapsto Spin (n)$ in the same way, since that requires killing two torsion groups ( $ℤ_{2}$ ). But I gather using Christoph’s point of view this is immediate:

As he points out, his 2-cocycle for $O (n)$ characterizes the extension $ℤ_{2} \to Spin (n) \to S O (n) .$ I suppose if we do a bit of “negative thinking” a la Toby Bartels we can tell a story how even one step further down his construction yields a 1-cocycle which characterizes $S O (n)$ with respect to $O (n)$ . Hm…

As Hisham Sati taught me, there is indication that for the purpose of physics we need to go up to the 11-connected cover of $Spin (n)$ given by the strict 10-group to be called $Ninebrane (n)$ . The NS-9-brane is the “space filling nine-brane at the Hořava-Witten-end-of-the-world” or the like, and one expects some curious properties, with its worldvolume theory being closely related to the target space theory of the heterotic string.

Hisham has some articles on the arXiv with tentative discussion of this point, and he tells me that some other group recently was able to make some aspects of these NS 9-branes more concrete.

In any case. To get to $Ninebrane (n)$ involves not just killing $π_{11} (O (n)) = ℤ$ but first also the two torsion groups $π_{8} (O (n)) = π_{9} (O (n)) = ℤ_{2}$ .

Posted by: Urs Schreiber on September 23, 2008 10:46 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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Urs wrote:

I still haven’t quite found the time to think about this in the detail that I ought to, but it seems that this is a way of looking at the construction of higher String-like extensions by successively killing homotopy groups.

Let me warn you: there’s a big difference between ‘killing’ homotopy groups and ‘cokilling’ cohomology groups. The first is fundamental to Postnikov towers. The second is fundamental to constructing the string group. I was confused for a long time about the difference!

To ‘kill’ a specific element $α$ of the $n$ th homotopy group of a space $X$ , we first pick a map from $S^{n}$ to $X$ representing $α$ . Then we glue an $(n + 1)$ -ball to $X$ using this map. We get a new space $\tilde{X}$ and an inclusion $X \to \tilde{X}$ . Pushing forward along this inclusion sends the element $α \in π_{n} (X)$ to $0 \in π_{n} (\tilde{X})$ .

To ‘cokill’ a specific element $β$ of the $n$ th cohomology group of a space $X$ , we first pick a map from $X$ to $K (ℤ, n)$ representing $β$ . Then we take the homotopy fiber of this map. We get a new space $\tilde{X}$ and a fibration $\tilde{X} \to X$ . Pulling back along this inclusion sends the element $β \in H^{n} (X)$ to $0 \in H^{n} (\tilde{X})$ .

These constructions are ‘dual’ in a certain sense. I could make the duality even more evident if I had used phrases like ‘homotopy cofiber’ and ‘cofibration’ when describing how to kill homotopy groups, to match my use of ‘homotopy fiber’ and ‘fibration’ when describing how to cokill cohomology groups. But that would make the simple process of gluing on a ball seem more scary!

To form the topological group $String (G)$ , we take our compact simply-connected simple Lie group $G$ and cokill the generator of its 3rd cohomology group. This has the side-effect of making its 3rd homotopy group trivial, but it’s not the same as killing the generator of the 3rd homotopy group. You can sense this by noting that we have a nice map $String (G) \to G$ , not $G \to String (G)$ .

I think some people say $String (G)$ is built by ‘killing the 3rd homotopy group’ of $G$ . I used to be guilty of this myself! But this is a sloppy usage, different from what homotopy theorists mean when they speak of ‘killing homotopy groups’ in the subject of Postnikov towers. So, somewhere around the time we wrote our paper on the string group, I started talking about ‘taking the homotopy fiber of the map $G \to K (ℤ,3)$ that represents the generator of the 3rd cohomology’. The point is not to sound more fancy: it’s to distinguish two different constructions with very different properties!

Posted by: John Baez on September 24, 2008 1:52 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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Some terminology, to help people who want to look up these things in, say, Hatcher’s freely available book.

The “cokilling” construction is called forming the $n$ -connected cover, since it generalizes the universal cover.

And the resulting tower of spaces is usually called the Whitehead tower.

Posted by: Dan Christensen on September 24, 2008 3:08 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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Thanks! I knew the ‘ $n$ -connected cover’ terminology, but forgot to mention it here. I hadn’t heard the term ‘Whitehead tower’.

Long time no see!

Posted by: John Baez on September 24, 2008 7:03 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Whitehead towers were mentioned in the thread on 2-covers. I see I was guilty of the killing/co-killing confusion.

Posted by: David Corfield on September 24, 2008 8:45 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Whitehead tower in the west and, dually, Postnikov tower in Moscow, were discovered about in the same period.
I think that I read in the volume Golden Year of Moscow Mathematics (Smilka Zdravkovska ed.) or elsewhere
about Whitehead being in Moscow visiting about the same time, but somehow there happened that Postnikov and Whitehead did not realized at the time that they were
working on a variant of the same idea. If somebody can check out the exact story…I do not have the book.

There are some other interesting historical points, e.g. the role of Pontrjagin, Aleksandrov and Kurosh in advising Postnikov what to do with his discovery. Pontrjagin was Postnikov’s advisor, otherwise helpful, but his advice was not crucial in estimating the importance of the construction, Aleksandrov did not really help, and Kurosh who was not a topologist but asked a right question.

Namely Postnikov devised his method for a particular computation, and Kurosh was asking about what else you can calculate with this, forcing the conversation to the point where Postnikov realized that knowing all “Postnikov invariants” in addition to the homotopy groups he can calculate EVERYTHING, hence that the whole set in fact determines the homotopy type.

Posted by: Zoran Skoda on October 7, 2008 5:44 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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Let me warn you: there’s a big difference between ‘killing’ homotopy groups and ‘cokilling’ cohomology groups. The first is fundamental to Postnikov towers. The second is fundamental to constructing the string group.

Thanks for the correction! (Or is it a rrection?)

I think some people say $String (G)$ is built by ‘killing the 3rd homotopy group’ of $G$ .

Yes. Not just some, it seems. Okay, better late than never… Thanks again.

Posted by: Urs Schreiber on September 25, 2008 11:46 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Is there a reason for the lack of correspondence between:

killing homotopy groups

and

cokilling cohomology groups?

You might have expected

cokilling cohomotopy groups,

but I guess cohomotopy has been given a different sense.

Posted by: David Corfield on September 25, 2008 12:16 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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Is there a reason for the lack of correspondence between:

killing homotopy groups

and

cokilling cohomology groups?

You might have expected

cokilling cohomotopy groups,

The Eckmann-Hilton dual of homotopy is cohomology, just as the Eckmann-Hilton dual of a sphere is a $K (Z, n)$ .

Up to (weak) homotopy, all spaces are built from spheres, and up to (weak) homotopy, all spaces are co-built from Eilenberg-Mac Lane spaces.

Eckmann-Hilton duality is discussed in Hatcher’s book, 4.H.

Posted by: Dan Christensen on September 26, 2008 12:43 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Thanks. Hatcher writes

There is a somewhat deeper duality between homotopy groups and cohomology, which one can see in the fact that cohomology groups are homotopy classes of maps into a space with a single nonzero homotopy group, while homotopy groups are homotopy classes of maps from a space with a single nonzero cohomology group. (p. 462).

The heuristic aspect to it is interesting,

Eckmann–Hilton duality can be extremely helpful as an organizational principle, reducing significantly what one has to remember, and providing valuable hints on how to proceed in various situations. To illustrate, let us consider what would happen if we dualized the notion of a Postnikov tower of principal fibrations, where a space is represented as an inverse limit of a sequence of fibers of maps to Eilenberg–MacLane spaces. In the dual representation, a space would be realized as a direct limit of a sequence of cofibers of maps from Moore spaces.

This entry notes that Eckmann-Hilton duality is

A duality principle variously described as “…a metamathematical principle that corresponding to a theorem there is a dual theorem (each of these dual theorems being proved separately)” [a4], “…a guiding principle to the homotopical foundations of algebraic topology…” [a1], “…a principle or yoga rather than a theorem” [a5], and “…a commonplace of experience among topologists, accepted as obvious” [a3]. The duality provides a categorical point of view for clarifying and unifying various aspects of pointed homotopy theory, but is often heuristic rather than strictly categorical.

Is it known when this strictness breaks down so that

It is not true, however, that dual theorems necessarily admit dual proofs,

and

It is also possible that the dual of a theorem is false?

Posted by: David Corfield on September 26, 2008 8:59 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Oh, my question over there under Math Miniatures
is answered here. What a tangled web we weave!

Posted by: jim stasheff on September 27, 2008 1:48 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Viro and Fuchs display the duality in two columns on pp. 17-20 of Topology II.

Posted by: David Corfield on September 26, 2008 9:49 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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David wrote:

Is there a reason for the lack of correspondence between

killing homotopy groups

and

cokilling cohomology groups?

There’s actually a perfect correspondence in everything except the terminology. Dan Christensen explained it, but perhaps a bit too tersely for the nonexpert. So:

The $n$ th homotopy group classifies maps from $S^{n}$ into a space, where $S^{n}$ is the space whose only interesting cohomology group is the $n$ th one, which is $ℤ$ .

The $n$ th cohomology group classifies maps from a space into $K (ℤ, n)$ , where $K (ℤ, n)$ is the space whose only interesting homotopy group is the $n$ th one, which is $ℤ$ .

Note that thanks to these two sentences, it follows that

“ $K (ℤ, n)$ is the space whose only interesting homotopy group is the $n$ th one, which is $ℤ$ .”

means the exact same thing as

“ $S^{n}$ is the space whose only interesting cohomology group is the $n$ th one, which is $ℤ$ .”

So, everything is perfect under heaven.

You might have expected

cokilling cohomotopy groups,

Or you might have expected “killing homology groups”. But those things both play other roles.

Posted by: John Baez on October 7, 2008 8:51 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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Are people interested in

$X$ is the space whose only interesting cohomotopy group is the $n$ th one,

$X$ is the space whose only interesting homology group is the $n$ th one?

Posted by: David Corfield on October 8, 2008 8:49 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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Above David was trying to understand why there is much ado about

$cohomology \leftrightarrow homotopy$

while at the same time nobody seems to care much about cohomotopy , much less about any duality of that to homology.

Experts will correct me, but I have recently come to think that the reason may be this:

cohomology and homotopy are both fundamental concepts, as follows: cohomology is about gluing and descent of presheaves, hence about $∞$ -stacks, whereas homotopy is about codescent of co-presheaves, hence $∞$ -costacks.

I have talked about that recently here.

Taking this as fundamental, homology in turn seems to be a rather contrived, rather derived concept: passing to homology is something like passing to the free symmetric monoidal completion of the archetypical $∞$ -costack living in homotopy theory: the fundamental $∞$ -groupoid. This free symmetric monoidal completio is something like the $∞$ -groupoid where all $k$ -cells of the fundamental $∞$ -groupoid may freely and invertibly be formally added. At the same time, homology proper forgets all the original composition operation in the fundamental $∞$ -groupoid.

So from this perspective it appears as a comparatively ad hoc, comparatively unnatural thing to consider. And this may help to reduce the surprise that its dual concept is of no big importance.

(Okay, that said, somebody should now mention the Eilenberg-Steenrod axioms and then I would greatly enjoy a discussion of the relation of cohomology in the sense of these axioms and cohomology in the sense of sheaf-cohomology/nonabelian cohomology etc.)

Posted by: Urs Schreiber on October 22, 2008 10:37 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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Urs,

You may be underestimating slightly the role of homology here.

There was an interesting example of an application of homology/cohomotopy some years ago when a problem of von Neumann was solved by Brown-Douglas-Fillmore. They showed that the obstruction to a positive answer was in a Steenrod K-homology group. This was further related to a cohomotopy group. Steenrod homology as such, or Steenrod-Sitnikov homology if you prefer, is a homology for compact spaces that, more or less, forms the system of Cech nerves of open covers, then takes chain complexes, then in addition forms their homotopy limit before taking homology.

This stuff then fed into the K-theory of operator algebras, and so back towards the quantum ideas more usually aired in this café.

I can search out references if anyone is interested.

Posted by: Tim Porter on October 23, 2008 9:45 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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You may be underestimating slightly the role of homology here.

That may be. At least my blatant statement triggered your interesting answer to David’s less blatant questions about the seeming lack of interesting dualities involving cohomotopy! :-)

There was an interesting example of an application of homology/cohomotopy some years ago

[…]

I can search out references if anyone is interested.

I’d be very much interested. Thanks!

Posted by: Urs Schreiber on October 23, 2008 10:33 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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There seems to be a discussion of this in the book on Analytic K-homology by Higson and Roe, but the original work was:

Extensions of $C^{*}$ -algebras, operators with compact self-commutators, and $K$ -homology,L. G. Brown, R. G. Douglas, and P. A. Fillmore, Bull. Amer. Math. Soc. Volume 79, Number 5 (1973), 973-978.

There is a substantial literature on this stuff. I think the point is that the cohomology of $C^{*}$ -algebras is related to the Steenrod type homology of the spectrum. But I am no functional analyst. The particular problem of von Neumann was to do with operators that were normal modulo the compact operators and the question was were they necessarily normal + compact. The answer was NO, the obstruction living in one of these homology type groups.

Posted by: Tim Porter on October 23, 2008 6:39 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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…cohomology and homotopy are both fundamental concepts, as follows: cohomology is about gluing and descent of presheaves, hence about $∞$ -stacks, whereas homotopy is about codescent of co-presheaves, hence $∞$ -costacks.

That raises a couple of questions. Is there a notion of co-gluing? And are cohomology and homotopy well named in view of their duality?

Posted by: David Corfield on October 23, 2008 9:46 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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Is there a notion of co-gluing?

Gluing is just another word for descent. So co-gluing is (or should be, I don’t think it is being used much at all so far) just another word for codescent.

I could have titled the entry Codescent and the van Kampen theorem simply Co-gluing.

And the story of the van Kampen theorem exhibts nicely the evolution of the ideas here. It’s quite nice:

let me say in the following “homotopy groupoid” for “fundamental $n$ -groupoid” (where $n$ could be some element in $ℕ$ or could be $∞$ depending on the precise setup) of a space (since that’s more evocative for our context and maybe would generally be a more suitable term alltogether).

Then: to understand the (vertex $n$ -groups of the) homotopy groupoid $Π (X)$ of a space $X$ which is covered $π : Y \to X$ by a space $Y$ , you can compute the homotopy groupoid $Π (Y)$ of $Y$ instead (which will be much simpler, for good choices of $Y$ ) and then co-glue that to obtain the groupoid $Codesc (Y, Π)$ .

If you do this for $n = 0$ in that you take $Π = P_{0} = Disc (-)$ the operation which sends a space to the disctete category over it, then the co-glued $Π$ -homotopy groupoid obtained from $Y$ is nothing but the familiar Čech groupoid $Codesc (Y, P_{0}) = Č (Y) = (Y^{[2]} \vec{\to} Y) .$

This is the weak pushout of $\begin{matrix} Y \times_{X} Y & \to^{π_{1}} & Y \\ ↓^{π_{2}} \\ Y \end{matrix}$ and is weakly equivalent to $X$ , which itself is the strict pushout. In fact, it is an acyclic fibration over $X$ $Č (Y) \overset{≃}{\to} > X .$

So we find that the category valued co-presheaf $P_{0} = Disc (-)$ is a co-stack: its value on any space is equivalent to the co-glued values on any cover of that space.

This story continues. When passing from $n = 0$ to $n = 1$ the above story becomes the story of the van Kampen theorem. Or rather: the van Kampen theorem is concerned with the strict pushout $\begin{matrix} Π_{1} (Y \times_{X} Y) & \to^{π_{1}} & Π_{1} (Y) \\ ↓^{π_{2}} \\ Π_{1} (Y) \end{matrix},$ whereas co-gluing of $Π_{1}$ s produces the weak pushout $Codesc (Y, Π_{1})$ . Again, we find that the weak co-glued pushout is an acyclic fibration over the strict pushout $Codesc (Y, Π_{1}) \overset{≃}{\to} > Π_{1} (X) .$ This is lemma 2.15, p. 15 in Parallel transport and functors.

So: the fundamental 1-groupoid, which is a groupoid-valued co-presheaf $Π_{1} : Spaces \to Groupoids$ is, too, a co-stack: its value on any space is equivalent to the co-glued values on any cover of that space.

And so on. The fundamental 2-groupoid co-presheaf $Π_{2} : Spaces \to 2 Groupoids$ is a 2-costack in that its value on any space is equivalent to the co-gluing of its value of any cover of this space. (This is appendix B here).

And are cohomology and homotopy well named in view of their duality?

Good question. Maybe not.

Posted by: Urs Schreiber on October 23, 2008 10:22 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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Can I be naughty and suggest that cohomology and homotopy are essentially the same thing?

Classical cohomology is homotopy classes of maps from X to some nice space or spectrum, non-Abelian cohomology is homotopy classes of maps from a space (or (hyper)covering of one) to a n-type. So cohomology is homotopy. In Pursuing Stacks, Grothendieck, if I remember rightly, talked about resolving the domain or coresolving the codomain i.e. the space that is the ‘coefficients’ in cohomology.

There is the slightly bizarre observation that Turaev introduced HQFTs as studying manifolds with extra structure given by a map to a ‘classifying space’. Brightwell and Turner introduced essentially the same construction interpreted as telling us information on the target ‘classifying space’. Both give valid, useful and interesting insights.

One may thus argue that both homotopy and cohomology are a question of comparing two objects, one known with one unknown. The difference is the position, domain or codomain, of the ‘unknown’. Sometimes they coincide!!!!!

Homology is more problematic and seems a lot less motivated, except when we come to homology of algebraic objects, where it can encode very useful behaviour.

Posted by: Tim Porter on October 24, 2008 6:06 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

both homotopy and cohomology are a question of comparing two objects, one known with one unknown.

as I recall, Bott and Tu point this out very clearly early on in their book

Posted by: jim stasheff on October 25, 2008 2:46 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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Can I be naughty and suggest that cohomology and homotopy are essentially the same thing?

Sure. But then allow me to be naughty, too, and insist that they are dual. And that the duality is simly that coming from passing from categories to opposite categories.

Classical cohomology is homotopy classes of maps from $X$ to some nice space or spectrum, non-Abelian cohomology is homotopy classes of maps from a space (or (hyper)covering of one) to a $n$ -type. So cohomology is homotopy.

Hm. To be frank, I don’t follow the conclusion here.

First I feel that “homotopy classes of maps” and “homotopy groups” is, while related, two different things. When I think of “cohomology versus homotopy” I am thinking of homotopy groups and their non-abelian generalizations.

And not all cohomology is invariant under homotopy! Of course everything following Eilenberg-Steenrod is (by definition), but there are perfectly respectable things called cohomology which are not. Sheaf cohomology in general is not! For instance Deligne cohomology is not. Even the differential versions of Eilenberg-Steenrod cohomology are not. In fact, whenever the cohomology classifies anythiing “with non-flat connection” it is not homotopy invariant.

(Of course there is the point of view that only generalized Eilenberg-Steenrod is “true” cohomology, by definition.)

So that summarizes my point of view currently. But I am emphasizing it mainly so that you can take it apart. If I am wrong, please tell me why.

resolving the domain or coresolving the codomain i.e. the space that is the ‘coefficients’ in cohomology.

Ah, i see. But wouldn’t there still be a difference then?

Cohomology is the colimit over (hyper)covers of maps out of these covers.

Homotopy should be limit over co-covers out of maps into co-covers.

This is dual. Not the same. No?

I may be mixed up. Please set me straight.

One may thus argue that both homotopy and cohomology are a question of comparing two objects, one known with one unknown. The difference is the position, domain or codomain, of the ‘unknown’.

but it’s still a difference, no?

Mayabe we are just using the words “the same” and “dual” differently, since it seems we perfectly agree on the technical aspects of the phenomenon.

Homology is more problematic and seems a lot less motivate

Ah, good. This seems to be the kind of statement I made above which got this discussion started: homology (and then cohomotopy) is somehow less fundamental than the pair cohomology and homotopy.

Posted by: Urs Schreiber on October 25, 2008 5:24 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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Of course, you are right if you want to interpret things in terms of homotopy groups. For me the homotopy groups are very poor invariants and the infinity cat models of homotopy types are much better, thus the 2-type of a space is not specified by the first two homotopy groups, but is by the crossed module (which only determines things up to coboundaries in a cohomology groups etc.) and so on.

The universal property of the fundamental group is not mirrored by the higher homotopy groups yet, if Grothendieck was right, algebraic models for n-types do have such a (lax coherent) universal property and that forms part of the basis of a lot of what we are trying to do.

Experience in other subject areas such as algebraic K-theory seems to show that one should consider the homotopy type as the central object of study with the groups and other structures, pairings, Whitehead products, etc, as being useful observations or probes of that homotopy type. Because of all this the complicated structure of sets of homotopy classes of maps between `things’ seems to be to be of more importance than homotopy groups or cohomology groups as they are `just’ particular instances of the overall situation. (I am not implying that those objects are not useful, nor that they are uninteresting, merely that for me they are less interesting.)

My reaction to your very valid points on Deligne cohomology, non-flat connections, etc., is that I always sort of expect some homotopy-like structure to be available, and it is the lack of geometric input to `homotopy’ that makes it inappropriate. Homotopy still enters sometimes via tools such as simplicial sheaves etc. and there the local information may use homotopy but the global does not.

My comment on Grothendieck’s resolving /coresolving quote was not checked up on. (I’m not sure where in the 650 pages of the manuscript it is !!!!!!)

You say: “Cohomology is the colimit over (hyper)covers of maps out of these covers.

Homotopy should be limit over co-covers out of maps into co-covers.”

The situation should probably be `homotopy limits’ rather than limits, but again it is a question of what are we to mean by `homotopy’, the homotopy type or some groups extracted using a `probing’ functor, such as $[S^{n}, -]$ .

Your talk of `co-covers’ seems to be related to Shape Theoretic ideas, where a space was studied using maps into a `nice’ space, in that case usually the Hilbert cube. (I can provide more info to anyone who is interested!!)

My problem with `classical homology’ is that the Moore spaces are a lot less related to group presentation type ideas than are Eilenberg-MacLane spaces so I end up a feeling of dissatisfaction. Cohomotopy often seems rather silly as there is no real reason for thinking that mapping into a sphere is a good thing to do. (Again that is a personal view, nothing more.) There is the historical use of these terms and they are not optimal from the perspective of this discussion.

There is one thought and that is that often in algebraic cohomology the coefficients are the same as the object being studied and you end up with $H_{*} (k, k)$ or similar, as being the key object of study. Then is this to be thought of as `homotopy’ like or cohomology like. My thought (possibly half baked) is that it is both and neither, but as I have never really studied these I should probably abstain from saying more!

My conclusion is that $[A, B]$ is probably more important to study in general than either $H^{*} (A, G)$ or $π_{*} (B)$ , and that really it is the homotopy types themselves, here of $A$ and $B$ , that need studying as directly as possible.

Posted by: Tim Porter on October 25, 2008 6:31 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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For me the homotopy groups are very poor invariants and the infinity cat models of homotopy types are much better, thus the 2-type of a space is not specified by the first two homotopy groups, but is by the crossed module (which only determines things up to coboundaries in a cohomology groups etc.) and so on.

So instead of the copresheaf $X \mapsto Π_{ω} (X)$ , the strict $∞$ -path groupoid, we need to eventually consider the weak fundamental $∞$ -groupoid. I suppose.

Posted by: Urs Schreiber on October 25, 2008 10:37 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

I think my point is that you divide out by the minimum for the calculations at that point (and that is partially a matter of taste!) The more baggage you carry around the harder it is to travel but the more likely it is that you have the right equipment when you need it! The problem is trying to optimise the balance point: what to carry v. what to dump. Sometimes I have noticed that initially you go through with an idea at a low level, only to realise that you can lift arguments to the next level up and possibly to all levels at once. That is so imprecise as to be useless perhaps, but dropping down too early to homotopy groups or even to infinity groupoids may destroy the geometric picture.

As an instance of this, you have been mentioning Ronnie and Phil Higgin’s crossed complex theory, which is a beautiful rich theory with a lot of lovely results which can probably be pushed further into the smooth case. There is however a relatively undeveloped theory of 2-crossed complexes which have slightly less Abelian information in them. They are related to Baues’s quadratic complexes. They have, instead of a crossed module at the base, a 2-crossed module, and so can encode a smidgen of the Whitehead product structure of a homotopy type. There is a corresponding infinity category model of course, with a weak interchange at the relevant level. Often it will be sufficient to work with crossed complexes, as the work involved is less arduous than for the next stage up. For general theory it would also be possible to use infinitely structured models with the possibility of Whitehead products etc in all dimensions. Possibly however, to get more calculations and more tractible theory, you may need to drop back to a finite level and it is the interplay of adjacent levels that also gives a lot of useful structural information.

Grothendieck mentioned early on in Pursuing Stacks that there would be as many models for n-types as there are mathematicians working with them as the exact choice would be a question of the individuals intuitions etc. I got the impression that he also meant that it would depend on the task the mathematician wanted that model to do. He accepted for instance that simplicial models for things were powerful but he did not like them probably because they were very large. What model suits in the smooth context is thus very difficult to gauge.

Posted by: Tim Porter on October 26, 2008 8:55 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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Just a very quick reply:

2-crossed complexes should be to groupoids in $(ω Cat, \otimes_{GrayCrans}) - Cat$ as crossed complexes are to groupoids in $ω$ -categories. Hence they know about the “first level” of non-strict composition.

Posted by: Urs Schreiber on October 26, 2008 11:05 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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That `feels’ right, but I would need to check details.

At the risk of repeating myself, I do think that the links between the $∞$ -Cat panoply and the machinery of Whitehead products etc. is an area that could, and perhaps should, be looked at in detail in the near future. The `Peiffer lifting’ map, (Conduché), does correspond to the Gray-tensor structure by old work of Joyal and Tierney, if I remember rightly, and it also links with the corresponding Whitehead product formula, but there I forget the exact formula. `Morally’ they are more or less the same in the relevant dimension (i.e. covering the interchange law.)

At the stack/costack level then the various homotopy operations ssuch as Whitehead products should be realisable by interesting pairing operations.

Posted by: Tim Porter on October 27, 2008 9:53 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

where do you see anything like Whitehead products in that infty-cat panolply?

Posted by: jim stasheff on October 27, 2008 2:15 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

The oldest weak infinity category model could be considered to be Kan complexes and thus simplicially enriched groupoids and there there are clear formulae for the Whitehead products as an ordered product of shuffle based terms given in an unpublished result of Kan. I do not know of a comparable result for other, say globular, models. (I have written out a proof of Kan’s formula if anyone is interested. The formula itself is on page 197 of the 1971 article of Curtis.)

Limiting the scope, in low dimsnional 2-type models (say Conduché’s 2-crossed modules), as I may have mentioned earlier, the Whitehead product can be linked to the Peiffer lifting and thus essentially to the lack of interchange in the corresponding categorical model. I seem to remember that Marco Grandis had some results in this direction, but am not sure. I know Ronnie Brown has a paper in which a formula linking the Peiffer lifting / h-map of a crossed square with the Whitehead product. (Again I can give a reference if I look hard enough or if I ask Ronnie!)

Slightly away from n-categorical models as such Baues introduced quadratic complexes and in his book on 4-dimensional complexes gave a formula involving Whitehead products, their definition being in his book on Algebraic Homotopy. Those quadratic complexes are modified versions of Conduché’s ones. The fact that his approach uses suspensions makes me wonder it John and Jim Dolan ever considered Whitehead products in their work.

My guess is that the Whitehead product formulae should be measure of lack of interchange at the various levels.

There was a talk given at one of the Como meetings by Florence Marty which looked at 3-categories following some ideas of Olivier Leroy. Her thesis title (1999) had been:

Approche en dimension supérieure des 3-catégories augmentées d’Olivier Leroy

I felt that some of the ideas she had put forward were possibly going in the direction of higher pairings related to Whitehead products, but I may be wrong… and perhaps someone else may know of this as well and can provide me with an an answer. There were various p + q interchanges evident in that stuff, and I am sure that in the other models for weak infinity cats there should be, either evident or lurking, similar structures.

Posted by: Tim Porter on October 27, 2008 3:46 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Tim wrote: My guess is that the Whitehead product formulae should be measure of lack of interchange at the various levels.

Samelson products seem more reasonable in
monoidal cats. Topologically Whitehead and Samelson are equivalent but in a cat context
would Whitehead obtsin at the nerve level?

Posted by: jim stasheff on October 27, 2008 6:06 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

Samelson products are more appropriate unless one is working with n-groupoid type objects. I was being sloppy in my terminology, sorry!

Does anyone know of other references for Samelson type products, say in monoidal categories as Jim suggests.

Posted by: Tim Porter on October 27, 2008 9:23 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

The paper of Ronnie’s is

74. “Computing homotopy types using crossed $n$-cubes of groups”, {\em Adams Memorial Symposium on Algebraic Topology}, Vol 1, edited N. Ray and G Walker, Cambridge University Press, 1992, 187-210.

The number refers to the numbering in his publication list at

http://www.bangor.ac.uk/~mas010/publicfull.htm

which contains a link to a pdf file of the paper. The result to note is his theorem 2.4 on page 199. The proof is given later in the paper.

Posted by: Tim Porter on October 27, 2008 6:07 PM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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or coresolving the codomain i.e. the space that is the ‘coefficients’ in cohomology.

At the risk of blowing my own horn, I once figured out how to define cohomology with coefficients in a simplicial topological group, by doing just this - leaving the space in question alone, but forming a resolution along the lines of Segal in “Cohomology of topological groups” 1970 Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69). I got some nice monad, but then lack of experience rose up and stopped me from doing anything interesting with it.

Posted by: David Roberts on October 28, 2008 1:54 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

resolution of ____ as ____?

Posted by: jim stasheff on October 28, 2008 3:58 AM | Permalink | Reply to this

Re: Group Cocycles and Simplices

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There is a monad on simplicial topological groups such that the unit of the monad makes a simplicial topological group a subgroup of a contractible one. The cosimplicial resolution of the simplicial topological group is made up of contractible simplicial topological groups.

Maybe I was fooling myself and got something uninteresting, but since $∞$ -things are popular here now, I thought it was worth dusting off. Certainly a smooth version would be interesting, as well as the links to $L_{∞}$ -algebras.

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September 22, 2008