# Classification of manifolds and smoothness structures

marcus
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Connes still relies on a differentiable (spin-) manifold structure, otherwise he would not be able to introduce a Dirac operator; in that sense his approach does not solve any of the above mentioned problems, as far as I can see
Tom, I think this is mistaken. Connes NCG setup does not rely on a manifold of any kind. In a spectral triple (H, A, D) the A is a * algebra of operators on a hilbertspace H and the D operator satisfies certain axioms. Nowhere is there a manifold.

It may seem as if Torsten thinks that a manifold is required, but his statement seems to me to be unclear. I would urge not to take his post as confirmation of the mistaken idea that a manifold is required in the definition of NCG.

Manifold IS involved in the definition of the special "almost commutative" case, where functions on a manifold are used in the construction. But we don't know that is is the necessarily the right way to go. There can be other ways to apply NCG. I doubt that "almost commutative" setup is the final word.

marcus, how would you define D, its spectrum and its heat kernel w/o a manifold

marcus
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marcus, how would you define D, its spectrum and its heat kernel w/o a manifold
You have to ask Connes. He defines it on stuff that does not consist simply of functions on manifold (that would be a *commutative*, he defines D on noncommutative structures too, sometimes on finite dimensional vectorspaces). D does not have to be anything you are used to. for him, D can just be something he spells out that satisfies certain axioms. The axioms that D satisfies make it perform the services that he needs, and allow in some commutative cases to RECOVER or reconstruct an algebra of functions defined on a manifold. So it is a generalization.

I'm not an expert, Tom. I just attended a Marc Rieffel seminar on this stuff for several weeks a few years back until his grad students began presenting papers I couldn't follow on obscure side topics, their own research etc etc. I don't remember much.

Maybe if this Marcolli Suijlekom thing develops I will go back and review one of the Connes papers defining NCG that gives the Spectral Triple axioms.

marcus
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I suppose the point of calling it "noncommutative" is that it is NON-MANIFOLD BASED. If you take a manifold and take the set of functions defined on it, that is an ALGEBRA because it is a vectorspace and you can multiply two functions together point wise. And multiplication commutes. So with minimal assumptions you have a commutative * algebra. And by looking at the structure of that algebra (the maximal ideals in it) you can recover something about the geometry.

So you are just doing geometry (but instead of dealing with the manifold itself you are "seeing" the manifold thru the algebraic structure of the * algebra of functions defined on it. Basic stuff from over 60 years ago--gelfand, naimark, segal?---I forget)

You are doing geometry but THRU a *commutative* algebra of functions that contain all the information about the manifold they are defined on.

So I think what Connes did was to axiomatize this and say "what if the algebra is not commutative?" What if the algebra is NOT the algebra of functions defined on a manifold, which would make it commutative, but something else? A more general idea of "geometry".

In the commutative case the functions defined on M tell you essentially all about M so you can do geometry without touching the manifold but only studying the algebraic structure. But if the algebra is not commutative then it does not come from a manifold and you can still study the "geometry" implied by the algebraic structure, but what is it the geometry OF??? Very strange. This was his creative idea. To generalize differential geometry to something else besides differential manifolds.

I have a lot of family things to do and don't have time to review and be sure about details. some of the others like brian or someone else would be better at explaining NCG anyway. I'll try again tomorrow if no one else volunteers.

arivero
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I suppose the point of calling it "noncommutative" is that it is NON-MANIFOLD BASED.
Well, sometimes you use NCG only because you want to manipulate a quotient of a manifold by a group action in a very generic way, including both manifold and non-manifold quotients. Or you want work out a groupoid equivalence relation. I am not sure of how useful NCG is if you only want to work with usual manifolds, surely it provides some new tools when it comes to calculate suspensions, borders, cohomologies, K-theory, and all these high math stuff.

arivero
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Except for the work of Torsten I am not aware of any paper which discusses these topics.

Some ideas:

5) we know that in dim = 4 there are in general uncountably many smoothness structures for a given (non-compact) manifold

5'') 11= 7 + 4

Exotic structures in dim=7 should be also of some interest.

julian
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In the restricted domain the canonical formulation is the better starting point. In the unrestricted domain no canonical formulation will exist - neither constructed directly, nor indirectly via the PI. I think topology and smoothness structures a the weak point of all approaches to QG I know - except for thorsten's work which I still do not really understand.
There is Thiemann's, Giesel's `Algebraic quantum gravity' [arXiv:gr-qc/0607099]: "While AQG is inspired by Loop Quantum Gravity (LQG), it differs drastically from it because in AQG there is fundamentally no topology or differential structure...The missing information about the topology and differential structure of the spacetime manifold as well as about the background metric to be approximated is supplied by coherent states."

Tom, I think this is mistaken. Connes NCG setup does not rely on a manifold of any kind. In a spectral triple (H, A, D) the A is a * algebra of operators on a hilbertspace H and the D operator satisfies certain axioms. Nowhere is there a manifold.

It may seem as if Torsten thinks that a manifold is required, but his statement seems to me to be unclear. I would urge not to take his post as confirmation of the mistaken idea that a manifold is required in the definition of NCG.

Manifold IS involved in the definition of the special "almost commutative" case, where functions on a manifold are used in the construction. But we don't know that is is the necessarily the right way to go. There can be other ways to apply NCG. I doubt that "almost commutative" setup is the final word.
Marcus, there is a fundamental misunderstanding. I spoke about Connes model to obtain the standard model of elementary particle physics (including the Higgs sector). There, he used a manifold (with a cross product of a discrete space), which must be spin and Riemannian.
You are of course right, in the original definition of a spectral triple, there is no manifold. But many examples of noncommutative spaces are derived from or directly related to manifolds. Take the leaf space of a foliation: one is unable to obtain any results of foliation theory by considering the funtion space over the leaf space. So, the introduction of an operator algebra instead of a (commutative) function algebra was an important step.
In the spectral triple (A,H,D), the Dirac operator D (usually called a self-adjoint operator) acts on the Hilbertspace H and A is the space of operators over H. A is a * algebra and the norm of the commutators [a,D] for alle elements a of A must be bounded.

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marcus
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Good point, this should give us some clue.
And then there are those approaches that point towards getting rid of formulations in terms of Lorentzian manifolds (a lá Connes non-commutative algebra) because they seem too limiting but I think those are not introducing gravity yet..
Connes still relies on a differentiable (spin-) manifold structure, otherwise he would not be able to introduce a Dirac operator; in that sense his approach does not solve any of the above mentioned problems, as far as I can see
You are right it doesn't solve the OP problems.
Maybe I'm confused, but does Connes really require this? I thought one was, at some level, just working with an algebra of non-commutative degrees of freedom plus extra data specifying the Dirac operator. This algebra indirectly determines a "non-commutative manifold" in the sense that the usual commutative algebra of functions plus extra data also determines a conventional commutative manifold.
In what I've read there is no smooth manifold required---instead there is a bunch of axioms that say (among other things) how the Dirac operator should behave.

In special cases/examples he can start with an algebra of functions on a manifold, so there is a manifold in the picture. But there does not have to be.

Tom may be thinking of a special case relevant to physics where the algebra is "almost commutative" in the sense that a big piece of it is constructed by taking the functions on a conventional manifold (and that part of the algebra is commutative) but then a little non-commutative piece is tacked on. The direct sum of the two is non-commutative. this construction comes up when they want to realize the standard particle model.
Marcus, there is a fundamental misunderstanding. I spoke about Connes model to obtain the standard model of elementary particle physics ...
I understand, Torsten. I was the one who brought up the special application of NCG which DOES involve a conventional 4d manifold in constructing the commutative part and which "obtains the standard model of elementary particle physics ..."

If you look back to the earlier posts you will see that the earlier references to NCG were not about the special application to standard model, but about NCG in general. It is important to make very clear that NCG does not require a manifold. It is merely one of the options to include a commutative (manifold-based) piece.

There was no "fundamental misunderstanding". Unless you thought that the discussion was just about the NCG standard model construction. In your post you did not make clear that you were talking about a specialized application of NCG, so Tom seems to have read what you said as applying to NCG in general.

The whole discussion was confusing, one spoke about gravity, Tom spoke about Dirac operator (as the self-adjoint operator) and all the others spoke about the general NCG.
My post was a direct reaction to Toms. I wanted to clarify that in Connes model using the Dirac operator (which is Connes version of the standard model, no other model was proposed by him) he assumed a Riemannian spin manifold.
I'm deeply sorry for the confusion. Next time I will retain myself in posting such things.

marcus
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The whole discussion was confusing, one spoke about gravity, Tom spoke about Dirac operator (as the self-adjoint operator) and all the others spoke about the general NCG.
My post was a direct reaction to Toms. I wanted to clarify that in Connes model using the Dirac operator (which is Connes version of the standard model, no other model was proposed by him) he assumed a Riemannian spin manifold.
I'm deeply sorry for the confusion. Next time I will retain myself in posting such things.
No problem, Torsten! I'm sorry for whatever confusion I contributed by not being clear. You surely know a lot more about non-manifold geometry than I do. I'm very excited by the Suijlekom-Marcolli paper which introduces the concept of a "gauge network", have you looked at the paper?

They are aiming at the Connes standard model without any commutative part of the algebra! All the spectral triples are purely non-manifold and also they are finite dimensional i.e. in A,H,D the algebra A is finite dimensional and so is the hilbertspace.

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I'm very excited by the Suijlekom-Marcolli paper which introduces the concept of a "gauge network", have you looked at the paper?

They are aiming at the Connes standard model without any commutative part of the algebra! All the spectral triples are purely non-manifold and also they are finite dimensional i.e. in A,H,D the algebra A is finite dimensional and so is the hilbertspace.
I remembered to read the paper in January but now I read it again. In the paper there was a clear motivation. Everything looks very interesting. But I'm not surprised that the authors obtained the Higgs sector. A spin network with a NCG at every vertex....
One thing troubles me: the spin network was usually constructed to be related to the triangulation of the 3-manifold. I know of only one approach to simplify a triangulatio of a 3-manifold: use a 2-complex (a so-called spine of a 3-manifold). It is known that every 3-manifold can be constructed by using spines but nothing less (including no 1-complex like a spin network).
But I have to think more carefully about it.

marcus
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... to simplify a triangulatio of a 3-manifold: use a 2-complex (a so-called spine of a 3-manifold). It is known that every 3-manifold can be constructed by using spines but nothing less (including no 1-complex like a spin network).
But I have to think more carefully about it.
Is a "spine" in this case the same as the 2-skeleton?
I think I know what you mean. The 2-skeleton is a 2-complex that simplifies the triangulation of a 3 -manifold by discarding the 3-cells and just leaving the vertices edges faces.

Torsten, I decided it would be more "on-topic" to reply to you in the "Dynamics of NCG" thread that PhysicsMonkey started as an offshoot of this one. So I quoted and replied to your post here:

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julian
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You cant classify 3 manifolds with 1d spin networks but what about q-deformed spin networks? Here vertices are replaced with discs and links with ribbons which can have twists in them...isn't this related to Dehn surgery and Kirby calculus or something? What is the additional information in q-deformed spin network states? Just the twists?

You cant classify 3 manifolds with 1d spin networks but what about q-deformed spin networks? Here vertices are replaced with discs and links with ribbons which can have twists in them...isn't this related to Dehn surgery and Kirby calculus or something? What is the additional information in q-deformed spin network states? Just the twists?
Hi Julian, it is not directly but related to Kirby calculus etc. q-deformed spin networks are more related to skein theory (as far as I can remember Kauffman proved that there is a strong relation). Skeins are related to knot theory and the knot polynomials. The relation to Kirby calculus was proved by Lickorish, he obtained 3-manifold invariants (of Witten type).

Sorry I forgot something to write:

Hi Julian, it is not directly but related to Kirby calculus. The q-deformed spin networks are more related to skein theory (as far as I can remember Kauffman proved that there is a strong relation, see the paper in Int. J. Mod. Phys. A 5 (1990) 93). Skeins are related to knot theory and the knot polynomials. The relation to Kirby calculus was proved by Lickorish, he obtained 3-manifold invariants (of Witten type), see J. Knot Theory and Its Ramifications, 2 (1993) 171.
In both cases the quantum group SU_q(2) was used.
Torsten

MTd2
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It seems some big shot guys from string theory uploaded a paper on exotic smoothness but did not cite Torsten. I feel really sad because I consider him a friend. It's also sad that they mentioned an expression contained in the title which is a famouls book on exotic smoothness, without citing it "wild world of 4 manifolds".

http://arxiv.org/pdf/1306.4320v1.pdf

Haelfix
yes, of course

yes, why not?

I know; nevertheless it may be interesting to take it seriously.

The interesting thing is that if you define the counting AND sum over ALL dimensions (!) then the sum is naturally peaked at dim=4 due to the uncountably many smoothness structures for non-compact 4-manifolds.
So if you go over the paper I linked too where they compute 2+1 pure Ads gravity with toric boundary conditions, they actually explicitly compute the sum over manifolds (and topological structures). What's fascinating is the choices they make on how to partition the sum.

B/c of the AdS/CFT correspondance they are able to bootstrap an answer and it seems to be self consistent on both sides of the duality. That's really cool, and as far as I know, has only occurred for Chern-Simmons theory exactly.

Anyway, what peaked my interest was that they did indeed include nontrivial diffeomorphisms in the sum (so called modular transformations), that they did arrange the sum as an expansion around classical solutions followed by loop corrections (where the coefficients need not be small) and that they explicitly did not include singular configurations of the geometry. That they then compute an answer that seems to agree with the CFT side is quite nontrivial and interesting, and at least motivates that these might be the correct choices for 4d.