Rosé and A-M: Geometrization of Quantum Mechanics

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  • #26
Kea
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Careful said:
I am questioning the PHYSICAL motivation for doing this ...
The quick and dirty answer:

Is the question what is mass? physical? I believe so. Or, as Grothendieck may have put it: if you can tell me what a metre is I will gladly talk to you. :smile:
 
  • #27
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Kea said:
The quick and dirty answer:
Is the question what is mass? physical? I believe so. Or, as Grothendieck may have put it: if you can tell me what a metre is I will gladly talk to you. :smile:
No, not at all :smile: ! But why do you not give the honor to Helge to answer my questions? I think my mathematical objections are serious enough.
 
  • #28
Careful said:
No, not at all :smile: ! But why do you not give the honor to Helge to answer my questions? I think my mathematical objections are serious enough.
Careful, thank you very much for the careful critique. You are right if this basics are wrong no further discussion is needed. I have to carefully discuss this with torsten and we will give a reply. Stay tuned.
 
  • #29
mike2 said:
Isn't there a requirement that the number of particles/singularites must effect the metric on the 4-MF so that the more mass there is, the more the spacetime metric curves? Isn't this what you have? Or have you only connected the QFT algebra with a 4-manifold?
The particles generate source-terms (engery-momentum-tensor) in the grav. field eq. and by this changing the curvature of space-time.
 
  • #30
garrett said:
This discussion started on another thread, but I thought it best to bring it over here:
Hi Helge, I want to get something straight that's confusing me. I'm still just learning this stuff. I have a question about what you say above, and from this quote from your paper:
Are you really saying these are the number of differential structures for ALL manifolds of these dimensions?
This does agree with the wikipedia entry:
http://en.wikipedia.org/wiki/Differential_structure" [Broken]
Is this what you're saying? Because I think it's either not true, there's some miscommunication, or I'm really messing up.
(And it was a friend who pointed out to me this was a potential problem with your paper, I was just bumbling around confused.)
The table you quote is only true for spheres. Except in n=4 the sphere is not known to have an infinite number of DS, though it might. As a counter example, I read in Brans latest paper that the number of DS is 1 for R^n when n>4.
Could you help clear this up? Or maybe you need to fix your paper? I do really like the main idea.
You are right this is missleading. The table list only examples for the different dims - what is possible - as an illustration - we have to fix that. The formulations seems to suggest that the table is true for all 4-MF - sure not! There are only few results about 4d. One is: for a big class of non-compact 4-MF (inclusive R4 ) there are infinite uncountable DS.
For compact MF (the case we are interessed) is little known, also for S4 as you noted. We consider in the paper sufficent non-trivial 4-MF (as mentioned in footnote [35] simply connected, compact 4-MF with rank of the 2. homology group > 2). torsten is the expert for this, i think he will post a more comprehensive comment.
 
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  • #31
marcus said:
"In the introduction we have shown that there is a close relation between the transition of the DS and a singular connection with 3D supports. Such connections are expressed by singular 1-forms with 3D supports."

I am struggling at the very beginning of understanding this. I am familiar with connection being expressed by a 1-form (with values in a somewhat arbitrarily chosen Lie algebra)----unless I am confusing something, this is very usual.

But there is a lot that is new here. I think of an earthquake and a "FAULT-LINE" which is actually a fault surface going deep into the earth, and I try to imagine a 4D analog.

So there is a 3D "FAULT" hypersurface. And somehow the change in DS is closely related to a connection (or a 1-form) defined on this 3D "fault". This 3D thing is the SUPPORT of the 1-form.

It is a set. And when you make the algebra, you are using what looks like it might be ordinary set operations, like UNION and INTERSECTION of these support sets.

It becomes very urgent for me to try to understand how the support set of the 1-form can, in some way, characterize the earthquake that happens when you go from one DeeEss to another DeeEss.

I am a slow learner, it may take days before my brain stops smoking and making sparks and begins to understand this idea of transition of DeeEss.
I will try to answer but again wait for torsten. The paper explains DS only in a algebraic way. There are other approches which are suited for analysis of DS. The most important is the h-cobordism technique (used by Smale in the h-cobordism-theorem for dim>4). You have two 4-MF M1, M2 and tune M1 to M2 - that process building a 5D-MF W were M1, M2 are boundaries. If M1, M2 are not diffeomorph (different DS) in W exists a non-tivial sub-MF - the Akbulut-cork A tuning the sub-4MF A1, A2. A1,2 are contractible. Thus only the boundaries are of interesst and (with Friedmann) the boundaries are homology-3-Spheres. Thats the singular 3dim-supports! That means, the important thing which makes M1, M2 different in DS can be trace back to the singular 3d-supports - thats your
"FAULT" of earthquakes. But dont think about a localized FAULT-LINE etc. The 3d-supports were the change of DS is "concentrated" can be free moved (but not removed) on the MF by diffeomorphisms. The DS is in this meaning a global property of the 4-MF, but the "core" of this property is concentrated on a 3d-MF! Im sure torsten will explain that again and much better.
 
  • #32
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Helge Rosé said:
I will try to answer but again wait for torsten. The paper explains DS only in a algebraic way. There are other approches which are suited for analysis of DS. The most important is the h-cobordism technique (used by Smale in the h-cobordism-theorem for dim>4). You have two 4-MF M1, M2 and tune M1 to M2 - that process building a 5D-MF W were M1, M2 are boundaries. If M1, M2 are not diffeomorph (different DS) in W exists a non-tivial sub-MF - the Akbulut-cork A tuning the sub-4MF A1, A2. A1,2 are contractible. Thus only the boundaries are of interesst and (with Friedmann) the boundaries are homology-3-Spheres. Thats the singular 3dim-supports! That means, the important thing which makes M1, M2 different in DS can be trace back to the singular 3d-supports - thats your
"FAULT" of earthquakes. But dont think about a localized FAULT-LINE etc. The 3d-supports were the change of DS is "concentrated" can be free moved (but not removed) on the MF by diffeomorphisms. The DS is in this meaning a global property of the 4-MF, but the "core" of this property is concentrated on a 3d-MF! Im sure torsten will explain that again and much better.
Here are some resources for grokking this fascinating post:
http://mathworld.wolfram.com/h-Cobordism.html" [Broken]

And http://mathworld.wolfram.com/h-CobordismTheorem.html" [Broken], which also mentions Smale's great proof of the Poincare conjecture in dimensions greater than four.

And here, for an extra treat, is http://www.math.ucdavis.edu/~tuffley/sammy/h-cobordism.html" [Broken]
 
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  • #34
marcus
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garrett said:
...
The wikipedia entry on differential structures is now fixed:
http://en.wikipedia.org/wiki/Differential_structure
thanks all,
I see that Helge wrote the Wiki entry on DeeEss, which John Baez later emended.

plodding along: selfAdjoint post mentions homotopy equivalence and this is defined in:
http://en.wikipedia.org/wiki/Homotopy

so an h-cobordism between M and M' is an ordinary cobordism W
with inclusion maps f and f'

with the extra proviso that f and f' are homotopy equivalences which means that there exist maps g and g' such that

g: W -> M
fg : W -> W is homtopic to the identity on W
gf : M -> M is homotopic to the identity on M

and likewise with primes on (f', g', M')
OK :smile: *plod, plod* and now the "h-Cobordism theorem" which I gather is due to Smale 1961 (?) says that if W (compact simplyconnected) is an h-Cobordism between M and M' (dim > 4) then M and M' are diffeomorphic and in fact W is diffeomorphic to MxI
that is Mx[0,1] the unit interval which was probably how we were imagining it to begin with. so it is one of those great theorems which reassure us that the world is not completely crazy, but is a little bit like what we expected

but only, dammit all, for dimension >4, and dimension = 4 is apparently NOT how we expected----so the world comes back to bite us in the ass, again as usual

this is mostly just to let the rest of you know that I am still alive
 
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  • #35
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See why the Poincare conjecture in dim 4 is important?
 
  • #36
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mmmm hmmm
agreed
 
  • #37
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I am still waiting ... but as a response to the above. No, I don't see why the solved Poincare conjecture (by Freedman in 1982) is important for physics. This conjecture basically says that any compact 4 manifold homotopic to the 4 sphere is homeomorphic to it which limits therefore the number of exotic differential (as well as topological) structures. Can someone explain me WHY one should be interested in compact 4 - D manifolds homotopic to the 4 -sphere anyway ?
 
  • #38
Kea
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Quoting from the conclusions of the Brans paper (reference 3):

"The example [of the Schwarzschild singularity and Kruskal coordinates] helps to illustrate that in General Relativity our understanding of the physical significance of a particular metric often undergoes an evolution as various coordinate representations are chosen. In this process, the topology and differentiable structure of the underlying manifold may well change. In other words, as a practical matter, the study of the completion of a locally given metric often involves the construction of the global manifold structure in the process."

Interestingly, Penrose had a great intuition for the importance of these modern methods before they were developed. See for example the book Techniques of Differential Topology in Relativity (1972) Soc. Indust. Appl. Math.
 
  • #39
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Helge Rosé said:
The particles generate source-terms (engery-momentum-tensor) in the grav. field eq. and by this changing the curvature of space-time.
So the addition of each new particle changes the DS and adds a new source-term to the GR equations and increases the curvature. But a given number of particles has a determined curvature of a given DS which may have many different metrics. So what this says is that the curvature is an intrincis property independent of metric? But I thought curvature was determined by the metric. What am I missing?

And the algebra among the many different DS's is a Hilbert space. So if a Hilbert Space exists, then there must simultaneously exist all these DS's and with it the various curvatures. So it would seem that if the zero point energy exists, then there is a Hilbert Space, and so there must be a superposition of DS's and with each a superposition of curvatures. So does the Hilbert space algebra of the DS's translate into a Hilbert Space algebra for the curvature/metrics? If so, it would seem that we now have a quantum theory of geometry/gravity, right?

PS. Are there both positive and negative curvature in this programme?
 
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  • #40
Mike2 said:
So the addition of each new particle changes the DS and adds a new source-term to the GR equations and increases the curvature. But a given number of particles has a determined curvature of a given DS which may have many different metrics. So what this says is that the curvature is an intrincis property independent of metric? But I thought curvature was determined by the metric. What am I missing?
If we only consider the 4-MF with its DS (mathematically) - then the DS is compatible with different metrics. But if you also demand the Einstein eq. and putting the DS as a source term in it then you get one metric as solution of E. eq. You may change that metric by diffeomorphisms but this does not change the physics: Einstein eq and DS are invariant wrt diffeomorphisms.
Mike2 said:
And the algebra among the many different DS's is a Hilbert space. So if a Hilbert Space exists, then there must simultaneously exist all these DS's and with it the various curvatures.
At the end of section 3:

The completion of the algebra [tex](\mathcal{T},Tr)[/tex] is a complex Hilbert space. By fixing the parameter [tex]\tau[/tex] of the algebra [tex]\mathcal{T}[/tex] to be [tex]\tau=1/2[/tex], the completion of [tex]\mathcal{T}[/tex] corresponds to the Fock space of fermions in quantum field theory (see [PlyRob:94], chapter 2). That is a remarkable result: The self-adjoint projectors [tex]e_{k}[/tex] generate the creation and annihilation operators of the fermions. That means, for [tex]\tau=1/2[/tex] the algebra [tex]\mathcal{T}[/tex] is the standard Clifford algebra of anti-commutative operators. For the case [tex]\tau\not=1/2[/tex], [tex]\mathcal{T}[/tex] extends the standard quantum field algebra to a Temperley Lieb algebra.


So you have superpositions, yes. (see eq. (19,20))
So it would seem that if the zero point energy exists, then there is a Hilbert Space, and so there must be a superposition of DS's and with each a superposition of curvatures. So does the Hilbert space algebra of the DS's translate into a Hilbert Space algebra for the curvature/metrics? If so, it would seem that we now have a quantum theory of geometry/gravity, right?
PS. Are there both positive and negative curvature in this programme?
Im not sure. The change of DS is expressed by change of connection - not curvature or metric.

The particles are the transition (the difference) between two DS. It is like in "Einsteins lift": the force I feel is because the accelerated frame or grav. field - accelerted frames and grav. field are equivalent.

In our case: curvature is because of matter or transition of DS - matter and DS-transition are equivalent.

So, a DS-Transition - or call it particle creator - modifies the curvature but as source term via the Einstein eq. At the moment, I dont know if there is a direct mapping between DS and curvature, so that the algebraic structure of DS maps into a algebra of curvatures.

Maybe some other can explain this better? (Torsten?)
 
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  • #41
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Kea said:
Quoting from the conclusions of the Brans paper (reference 3):
"The example [of the Schwarzschild singularity and Kruskal coordinates] helps to illustrate that in General Relativity our understanding of the physical significance of a particular metric often undergoes an evolution as various coordinate representations are chosen. In this process, the topology and differentiable structure of the underlying manifold may well change. In other words, as a practical matter, the study of the completion of a locally given metric often involves the construction of the global manifold structure in the process."
Interestingly, Penrose had a great intuition for the importance of these modern methods before they were developed. See for example the book Techniques of Differential Topology in Relativity (1972) Soc. Indust. Appl. Math.
A change of differentiable structure when going from Schwarzschild to Kruskal coordinates, mmmm ?? That is not how you should see it: the manifold for the Schwartzschild differentiable structure does not contain the event horizon while the manifold for the Kruskal coordinates does (on the overlap, both differentiable structures are perfectly compatible). However, nothing physical is involved here (and I guess nothing physical happens in Helge's paper either) !!! The physical interpretation on the black hole horizon can be equally made using the Schwarzschild coordinates by taking suitable limits of the metric invariants towards the cut-out horizon. Anyway, I pointed out that albeit the pull back of the connection on N is a singular connection on M (which is invariant under coordinate transformations on N); the splitting they make in a ``regular´´ and ``singular´´ part is not intrinsic (with respect to N) at all !
 
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  • #42
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garrett said:
This discussion started on another thread, but I thought it best to bring it over here:
Hi Helge, I want to get something straight that's confusing me. I'm still just learning this stuff. I have a question about what you say above, and from this quote from your paper:
Are you really saying these are the number of differential structures for ALL manifolds of these dimensions?
This does agree with the wikipedia entry:
http://en.wikipedia.org/wiki/Differential_structure" [Broken]
Is this what you're saying? Because I think it's either not true, there's some miscommunication, or I'm really messing up.
(And it was a friend who pointed out to me this was a potential problem with your paper, I was just bumbling around confused.)
The table you quote is only true for spheres. Except in n=4 the sphere is not known to have an infinite number of DS, though it might. As a counter example, I read in Brans latest paper that the number of DS is 1 for R^n when n>4.
Could you help clear this up? Or maybe you need to fix your paper? I do really like the main idea.
Hallo, I'm the second author Torsten and try to answer your question.
At first one has to divide the manifolds into 2 classes: compact and non-compact. The table in our paper is only true for compact n-manifolds. In the non-compact case the number of structures may differ. For example, all trivial R^n have only one differential structure for [TEX]n \neq 4[/TEX]. The case n<4 is more or less trivial. The higher-dimensional case n>4 was covererd by Stallings 1962 by using the non-compact version of the h-cobordism theorem, the so-called engulfing theorem. By using the h-cobordism theorem, Kervaire and Milnor are able (around 1963) to classify the exotic spheres in dimension n>4. In a serie of papers, Kirby and Siebenman (in the 1970's) extend the work of Lashof, Mazur, Hirsch etc. to show that the number of differential structures of a compact manifold is the same as for the corresponding sphere. Thus, our table is true for all compact manifolds. The case n=4 is open for some "trivial" compact manifolds like [TEX]S^4,S^2\times S^2,{\mathbb C}P^2,...[/TEX]. It is verified for more complex manifolds like the K3 surface.
I hope that will resolve your confusion.
 
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  • #43
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Careful said:
I am still waiting ... but as a response to the above. No, I don't see why the solved Poincare conjecture (by Freedman in 1982) is important for physics. This conjecture basically says that any compact 4 manifold homotopic to the 4 sphere is homeomorphic to it which limits therefore the number of exotic differential (as well as topological) structures. Can someone explain me WHY one should be interested in compact 4 - D manifolds homotopic to the 4 -sphere anyway ?
Hi, before I react on your other remarks, I will say some words about the Poincare conjecture in dimesnion 4:
Freedman proved in 1982 that any manifold which is homotopic to the 4-sphere then this manifold is homeomorphic to the 4-sphere. As Donaldson showed by a counterexample, the smooth variant of this theorem breaks in dimension 4. Thus, there is the possibility that the 4-sphere has an infinite number of differential structures. By the so-called Gluck construction such possible candidates were constructed but a suitable invariant is missed to distinguish them.
 
  • #44
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torsten said:
Hi, before I react on your other remarks, I will say some words about the Poincare conjecture in dimesnion 4:
Freedman proved in 1982 that any manifold which is homotopic to the 4-sphere then this manifold is homeomorphic to the 4-sphere. As Donaldson showed by a counterexample, the smooth variant of this theorem breaks in dimension 4. Thus, there is the possibility that the 4-sphere has an infinite number of differential structures. By the so-called Gluck construction such possible candidates were constructed but a suitable invariant is missed to distinguish them.
I never claimed otherwise! I simply said that this theorem implies that the number of differentiable structures on any topological compact four manifold homotopic to the 4 sphere is the same as that for the topological four sphere itself (meaning that homotopy is not going to add any other forms of exotism).
 
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  • #45
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Careful said:
Hi, I got to page five and have already loads of technical questions/remarks. The authors start by noticing that a differentiable structure carries lot's of topological information and provides as well the necessary mathematical setting to write out the Einstein Field equations. That is certainly correct, ONE differentiable structure actually determines all Betti numbers (by studying critical points of vectorfields). However, the authors are not pleased with the knowledge of the number of multidimensional handles and want to include exotic differentiable structures associated to a topological manifold. Any good motivation for this is lacking; string theorists would actually jump out of the roof since in ten dimensions, only six inequivalent differential structures exist. It would be instructive to UNDERSTAND why in dim 2 and 3 (one is easy to proof) only one differentiable structure exists and what makes four so special, but no such insight is provided. For example: one should know if an explicit algorithm exists for creating such inequivalent types. The authors do suggest in that respect the use of surjective, smooth (between two inequivalent differentiable structures) but not injective mappings, but this is by far not sufficient.
That is difficult to answer and hopefully the following is not to technical. In dimension 2 and 3 the uniqueness of the differential structure can be shown where the problem is attributed to the 1-dimensional case. In 1982 Freedman classifies all topological, simply-connected manifolds to show that this classification mimics the higher-dimensional case. Thus, it is better to look at the higher-dimensional classification of differential structures by using the h-cobordism theorem. The failure of the smooth h-cobordism (Donaldson, 1987) opens the way to show that there are more than one possible differential structure on simply-connected 4-manifolds. For sufficient complicated 4-manifolds there is an explicite construction by Fintushel and Stern using knots and links (see the pages 9 and 10 of our paper for a description). Now, why is dimension 4 so special? The interior of a h-cobordism between two topologically equivalent 4-manifolds M,N consists of 2-/3-handle pairs. All other handles can be killed by using Morse theory (see Milnor, Lectrures on the h-cobordism theorem). These 2-/3-handle pairs can be killed if and only if there is a special embedded disk (the Whitney Disk). But if the disk has self-intersection then this disk ist not embedded. But that happens in dimension 4 by dimensional reasons. In higher dimensions there is no self-intersections and thus such a Whitney disk always exists. In our paper we use these results to consider a smooth map f:M->N between M,N with different differential structures. The singular set is by definition the set where the map f is not invertable or where df=0. So, we start with a mathematically given situation: two topological equivalent 4-manifolds with different differential structures. In the paper we are not dealing with the question to decide wether two 4-manifolds are diffeomorphic or not. That question has to be addressed later.

Careful said:
Some technical comments regarding section II: the definition of a singular set is very strange, one would expect df_x to have rank < 4 and not df_x=0.
Section III deals with pulling back the tangent structures from a differentiable structure N to a differentiable structure M by a mapping f. The authors define the singular CONNECTION one form G associated to f There is not given any rigorous definition of G = f_{*}^{-1} d f_{*} since this expression is meaningless where df_x has rank < 4 (since
f_{*}^{-1} does not exist there), so at least one should do this in the distributional sense wrt to a volume form determined by an atlas in the differentiable structure.
That is correct. The paper is written for physicists and we are not dealing with the theory of currents which is necessary to understand such singular objects. I know that problem. The whole idea based on the work of Harvey and Lawson about singular bundle maps. The map df:TM->TN is such a singular bundle map and Harvey and Lawson define a current Div(df) which they call a divisor. It is true that the expression in the given form as G = f_{*}^{-1} d f_{*} is mathematically not rigorous but it can be defined by the theory of Harvey and Lawson (see HL, A theory of characteristic currents associated with a singular connection, Asterisque 213, 1993). In the following we consider the cohomology class associated to the current (in the sense of Federer, Geometric measure theory). Thus, a diffeomorphism of M AND/OR N does not change this class. Secondly, by a result of Freedman, two homotopy-equivalent 4-manifolds are homeomorphic. Thus, the cohomology classes are connected to the differential strcuture. That agrees also with the results of Seiberg-Witten theory where special cohomology classes (called basic classes) determine the differential structure. What I want to say is that one can give the experssion G a sense by using the theory of Harvey and Lawson. I see that we forgot to write it.
Careful said:
A second comment is that G is not anything intrinsic - it is just a (distributional) gauge term and NOT a one form. Therefore, it is an uninteresting object related to a specific mapping f and to a choice of coordinate systems on M AND N (and especially this last property is very bad) - admittedly, it depends slightly upon the change of differentiable structure (through f) and does give rise to a distributional source in the energy momentum tensor. Nevertheless, the authors want to do something with it and give two inequivalent definitions for G; one based on nontrivial connections and one on the flat connection.
Yes it is right that the G in that form depends on the differential structure. That is the reason why we take the trace of the connection or curvature to exclude the dependence of the diffeomorpism. We always use in the paper the fact that a cohomology class can be associated to a current and vice versa.
Careful said:
The definition of the support is fine (since one wants to single out the singular part). With the definition of the product, something strange happens: the authors seem to consider G as a ONE FORM (which it isn't) and POSTULATE that the singular support of G is a three manifold and want to associate a specific generator of the first fundamental group to it. Poincare duality as far as I know is a duality between cell complexes of dimension k and n-k or homology classes of dimensions k and n-k, and this is clearly not the case. What the authors seem to allude to is the duality between the first homotopy class and the first homology class, which is the de Rahm duality and this could be only appropriate in case the singular support of G is a three manifold but still there is NO CANONICAL ONE FORM given, which is the other essential part of de Rahm theory. The same comment applies to the use Seifert theory; this COULD be only meaningful when the singular support of G is a three manifold, which is NOT necessarily the case (for a generic surjective, non injective, smooth f, the singular support could not even be a manifold) - the authors should provide a theorem that this is so. The latter is necessary since the theory of knots makes only sense in three dimensions (and M is a four dimensional manifold).
I think these issues need clarification otherwhise it seems to go wrong from the beginning...
It is not necessary to consider G as a one form. You can also consider G as a current with support [tex]\Sigma[/tex]. Let f:M->N be a singular map. Now I will say some words about the structure of [tex]\Sigma[/tex]. For that purpose, we have to say some words about the theory of singular maps. In topology, we are only interested in such topological characteristics like intersection points which are coupled to the question when two sub-manifolds intersect transversal. In most cases that happens and
we are done, but according to Sard's theorem a smooth mapping [tex]f:X\to Y[/tex] between smooth manifolds [tex]X,Y[/tex] has a set of critical
values of f of measure zero. That means, there are some (countable
many) cases where we don't get a transversal intersection between
sub-manifolds represented by some map [tex]f:X\to Y[/tex]. The question is
now: which deformation of the smooth map f to [tex]\tilde{f}[/tex] given by
a deformation of the smooth manifolds [tex]X,Y[/tex] eliminates the critical
(or singular) values of f. Such a procedure is called unfolding of f and Hironaka proves the general theorem that for every singular map f there is a
sequence of operations which unfolds f. These operations are
usually called blow-up and blow-down. In our case [tex]f:M\to M'[/tex] a
blow-up leads to a map [tex]\tilde{f}:M\#{\mathbb C}P^2\to M'[/tex] and a
blow-down to [tex]\tilde{f}:M\#\overline{{\mathbb C}P}^2\to M'[/tex].
Hironakas theorem means that the unfolding of f leads to a
diffeomorphism
[tex]
\tilde{f}:M\underbrace{\#{\mathbb C}P^2\#\cdots\#{\mathbb C}P^2}_n\#
\underbrace{\overline{{\mathbb C}P}^2\cdots\#\overline{{\mathbb
C}P}^2}_m \to M'
[/tex]
and by using the diffeomorphism (see Kirby, Topology of 4-manifolds)
[tex]
(S^2\times S^2)\#{\mathbb C}P^2={\mathbb C}P^2\#\overline{{\mathbb
C}P}^2\#{\mathbb C}P^2
[/tex]
we obtain a diffeomorphism
[tex]
\tilde{f}:M\underbrace{\#S^2\times S^2\cdots\#S^2\times S^2}_m\#
\underbrace{\#{\mathbb C}P^2\#\cdots\#{\mathbb C}P^2}_{n-m}\to M'
[/tex]
where we assume w.l.o.g. [tex]m<n[/tex]. But that is nothing than a weaker
version of the famous theorem of Wall about diffeomorphisms between
4-manifolds (see Kirby). A very important concept is the
stable mapping. Let [tex]f\in C^\infty(X,Y)[/tex] be a smooth mapping [tex]f:X\to
Y[/tex]. Then f is stable if there is a neighborhood [tex]W_f[/tex] of f in
[tex]C^\infty(X,Y)[/tex] (we use the compact-open topology for that space)
such that each [tex]f'[/tex] in [tex]W_f[/tex] is equivalent to f. According to
Mather stable smooth mappings between 4-manifolds are
dense in the set of smooth mappings. Thus according to Stingley
(see the phd thesis under supervisition of Lawson) one has to focus on that particular subset to study
maps between homeomorphic but non-diffeomorphic 4-manifolds.
Locally such maps are given by stable maps between [tex]{\mathbb
R}^4\to{\mathbb R}^4[/tex], where there is two types: 2 maps (rank 2
singularities) with a 2-dimensional singular subset and 5 maps
(Morin singularities or rank 3 singularities) with a 3-dimensional
singular subset. Stingley extends this result to
smooth 4-manifolds and shows that the rank 2 singularities can be
killed by an isotopy for maps [tex]f:M\to M'[/tex] between two homeomorphic
but non-diffeomorphic 4-manifolds. Thus we are left with the rank 3 singularities. Furthermore the corresponding manifold is closed. That supports the use of Seifert theory.
That agrees with a result of Freedman, Hsiang and Stong. They analyse the failure of the smooth h-cobordism and prove a structure theorem. Then the h-cobordism can be divided into two parts: a trivial h-cobordism inducing the homeomorphism between the two manifolds and a subcobordism between two contractable submanifolds A1, A2 of M and N, respectively. The boundary of this submanifolds A1,A2 are homology 3-spheres (see Freedman, 1982). By the usual association between critical points of Morse functions and cobordism, it was shown (I forgot the reference, maybe Milnor) that a singular map and the cobordism are associated to each other.

Some words about the Poincare duality. Yes you are right. I use a combination of the Poincare duality to relate the k form to an n-k cycle. Then I use the duality of an k cycle and an n-k cycle for a compact manifold. The element of the fundamental group is related to homology class by using the Hurewicz isomorphism, i.e. I can only relate the elements of the fundamental group which are not belong to the commutator subgroup.
Hopefully you are satisfied with that explaination. Otherwise please write.
 
  • #46
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** In our paper we use these results to consider a smooth map f:M->N between M,N with different differential structures. The singular set is by definition the set where the map f is not invertable or where df=0. **

?? Not invertible means rank df_x < 4 and not necessarily df_x = 0.

**I know that problem. The whole idea based on the work of Harvey and Lawson about singular bundle maps. The map df:TM->TN is such a singular bundle map and Harvey and Lawson define a current Div(df) which they call a divisor. **

And you can define this current without introducing a background metric on M (I do not believe that) ?? Please give this definition (I do not have easy acces to the book of Federer).

** It is true that the expression in the given form as G = f_{*}^{-1} d f_{*} is mathematically not rigorous but it can be defined by the theory of Harvey and Lawson (see HL, A theory of characteristic currents associated with a singular connection, Asterisque 213, 1993). **

Why even bother defining G since it depends upon a choice of coordinates on N anyway ? Shouldn't one concentrate on the pull back of the covariant derivative ?

** In the following we consider the cohomology class associated to the current (in the sense of Federer, Geometric measure theory). Thus, a diffeomorphism of M AND/OR N does not change this class. **

Sure but you still need to tell me how to define the current.

**What I want to say is that one can give the experssion G a sense by using the theory of Harvey and Lawson. I see that we forgot to write it. **

But still G depends on the particular coordinate chart in N, even if I trace it in M (that is actually easily seen on the regular part of G- I do not even need to bother about the singular part). The rest of the message sounds acceptable (though I did not know many of these details).

Cheers,

Careful
 
Last edited:
  • #47
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Kea said:
People may be interested in an old thread on the spin foam connection
http://www.lns.cornell.edu/spr/2003-10/msg0055272.html
Hi, I'm the second author of the paper. Yes I find that work interesting. The spin foam approach is not so far away from our approach. For instance, Rovelli and Pietri showed by using Recoupling theory that the scalar product of the Loop quantum gravity is the trace of the Temperley-Lieb algebra.
Furthermore, one can remark that a PL structure in 4 dimensions is equivalent to a DIFF structure. That means that two non-diffeomorphic, but homeomorphic 4-manifolds differ also by the combinatorical structure. Thus, the spin-foam model of a 4-manifold describes the differential structure. But I think I don't tell anything new. (see Pfeiffers paper)
 
  • #48
Kea
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  • #49
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Careful said:
** In our paper we use these results to consider a smooth map f:M->N between M,N with different differential structures. The singular set is by definition the set where the map f is not invertable or where df=0. **
?? Not invertible means rank df_x < 4 and not necessarily df_x = 0.
**I know that problem. The whole idea based on the work of Harvey and Lawson about singular bundle maps. The map df:TM->TN is such a singular bundle map and Harvey and Lawson define a current Div(df) which they call a divisor. **
And you can define this current without introducing a background metric on M (I do not believe that) ?? Please give this definition (I do not have easy acces to the book of Federer).
** It is true that the expression in the given form as G = f_{*}^{-1} d f_{*} is mathematically not rigorous but it can be defined by the theory of Harvey and Lawson (see HL, A theory of characteristic currents associated with a singular connection, Asterisque 213, 1993). **
Why even bother defining G since it depends upon a choice of coordinates on N anyway ? Shouldn't one concentrate on the pull back of the covariant derivative ?
** In the following we consider the cohomology class associated to the current (in the sense of Federer, Geometric measure theory). Thus, a diffeomorphism of M AND/OR N does not change this class. **
Sure but you still need to tell me how to define the current.
**What I want to say is that one can give the experssion G a sense by using the theory of Harvey and Lawson. I see that we forgot to write it. **
But still G depends on the particular coordinate chart in N, even if I trace it in M (that is actually easily seen on the regular part of G- I do not even need to bother about the singular part). The rest of the message sounds acceptable (though I did not know many of these details).
Cheers,
Careful
OK I see the point. :smile: You are right. The pure definition of the current needs a metric for M and N but the definition of the sum and product don't depend on the particular metric. The intersection between sets and the linking of the curves don't depend on the metric.

But I have also a question: Why do you think that the differential structure on a 4-manifold has nothing to do with physics?
I think it is interesting for you that we are able to derive the Temperley-Lieb algebra by using the h-cobordism of 4-manifolds and the theory of Casson handles. The connection approach is not the only way to quantum mechanics.
 
  • #50
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Kea said:
Hi Torsten
I am very pleased to meet you. I am one of the authors of
http://www.arxiv.org/abs/gr-qc/0306079
:smile:
Hi Kea,
I am also very pleased to meet you. I think I know your work and find it very interesting. It remembers me on a construction in singularity, called the cone of a singularity. In that construction, the singularities of a function look like a cone. Thus, the change of a 3-manifold (visualized as a 4-dimensional cobordism) cane be visualized as a conical singularity of some function (related to the Morse function of the cobordism). The resolution of the singularity should end with a smooth 4-manifold but with non-trivial topology.
Unfortunately, I can't fill in all details.

But maybe more later

Torsten
:smile:
 

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