Rosé and A-M: Geometrization of Quantum Mechanics

[marcus]

Asselmeyer-Maluga and Rosé: Geometrization of Quantum Mechanics

this paper was mentioned by selfAdjoint in another thread.
people there seemed to think it should be studied/discussed
so maybe this paper should have its own thread, besides
being included in our list of new QG/matter ideas

http://arxiv.org/abs/gr-qc/0511089
Differential Structures - the Geometrization of Quantum Mechanics
Torsten Asselmeyer-Maluga, Helge Rosé
13 pages, 2 figures
"The usual quantization of a classical space-time field does not touch the non-geometrical character of quantum mechanics. We believe that the deep problems of unification of general relativity and quantum mechanics are rooted in this poor understanding of the geometrical character of quantum mechanics. In Einstein's theory gravitation is expressed by geometry of space-time, and the solutions of the field equation are invariant w.r.t. a certain equivalence class of reference frames. This class can be characterized by the differential structure of space-time. We will show that matter is the transition between reference frames that belong to different differential structures, that the set of transitions of the differential structure is given by a Temperley-Lieb algebra which is extensible to a C*-algebra comprising the field operator algebra of quantum mechanics and that the state space of quantum mechanics is the linear space of the differential structures. Furthermore we are able to explain the appearance of the complex numbers in quantum theory. The strong relation to Loop Quantum Gravity is discussed in conclusion."

my comment: this looks interesting. I would not have caught it. selfAdjoint flagged it.
https://www.physicsforums.com/showthread.php?p=834906#post834906
in post #7 of the Garrett Lisi thread.

what is impressing me most is that right now seems to be a time of new ideas. a lot of new ideas are appearing that connect different mathematical pictures of spacetime, all having to do with Quantum Gravity

 

[Kea]

initial reaction

Thanks selfAdjoint and Marcus

Well I guess you know what I'm going to say! All roads lead...

The references to Krol on page 2 are interesting. A recent and related paper by Krol is

Model Theory and the AdS/CFT correspondence
http://arxiv.org/PS_cache/hep-th/pdf/0506/0506003.pdf

Quote from the abstract: ...though explicit calculations refer to the would be noncompact smooth 4-invariants based on the intuitionistic logic.

I think it is just great that Rose' and A-M have spelled this out carefully.

 

[selfAdjoint]

what is impressing me most is that right now seems to be a time of new ideas. a lot of new ideas are appearing that connect different mathematical pictures of spacetime, all having to do with Quantum Gravity

Yes, I am excited about these new deep results. For example Asselmeyer and Rose' show that their approach restricts the underlying coefficient module to be the complex numbers, which would answer that puzzled about quantum physics. Schroedinger brough complex numbers in from nineteenth century theoretical optics, but they were never shown to be required before, AFAIK.

 

[Kea]

Helge is here! Hi!

 

[Kea]

Remember

Quantum general relativity and the classification of smooth manifolds
Hendryk Pfeiffer
http://arxiv.org/abs/gr-qc/0404088

Might be useful.

 

[marcus]

these two people are at what I think is a semi-private contract Research and Development organization. the byline says Fraunhofer- Gesellschaft

http://www.fraunhofer.de/fhg/EN/company/index.jsp
http://fraunhofer-society.biography.ms/the byline says FIRST FhG, Berlin.
FIRST must be an acronym for some department at Fraunhofer

Yes, FIRST means FRAUNHOFER INSTITUTE COMPTERARCHITECTURE SOFTWARE TECHNOLOGY

we would say FICST, but for them a computer is a "Rechner" (because it Reckons stuff) and so they say FIRST. I was wondering.
http://www.first.fraunhofer.de/

Maybe it is like being at an IBM Lab.

the FhG centers----there are many all over Europe----do CONTRACT research for both private companies and governments, they say they are the biggest organization for APPLIED research in Europe

Anyway these two young people Helge Rosé and Torsten A-M must likely be
BEGINNING researchers, because i don't find many previous papers by them,
only I think one by Torsten.
[EDIT: with Kea's help I found more papers by Torsten, he is more senior, has been working in this field 10 years, co-authored with Brans, is writing a book]

Well I didnt know about the Fraunhofer Institutes. You learn something new everyday.

I guess we all know about the famous Fraunhofer who was born in 1787 and invented spectroscopy---what much of atomic physics and astronomy is based on.
http://www.biography.ms/Joseph_von_Fraunhofer.html
It says he was orphaned at age 11, in 1798, so he had to go to work in a workshop, which however collapsed in 1801. Therefore as a young 14-year old Fraunhofer was buried in the remains of a badly constructed Munich lens-grinding factory. This however worked to his advantage, since he was rescued by the Prince of Bavaria who later became Maximilian Joseph the King of Bavaria. This prince was leading the crew digging people out, and he later helped Fraunhofer get time and books to study physics.

 

[marcus]

Helge is here! Hi!

yeah, I saw Helge was online here at PF, so that was what prompted me to start this thread. but it was sA who twigged the paper

here are snapshots of the two guys who wrote the paper, Torsten and Helge
http://mmm.first.fraunhofer.de/de/team/

 

[Kea]

Anyway these two young people Helge Rosé and Torsten Asselmeyer-M ...

Actually, it seems that Torsten has only recently appended the second part of his surname. See:

http://www.arxiv.org/find/grp_physi.../1/0/all/0/1?skip=0&query_id=884d475087264912



and also, one of the references in the paper

Generation of Source Terms in General Relativity by differential structures
T. Asselmeyer
14 pages
http://www.arxiv.org/abs/gr-qc/9610009

 

[Mike2]

Are these differential structure related to topologial quantum field theories? Quantum Mechanics is derived from Quantum field theory where we have creation and anhiliation operators for particles. The double slit experiment tell us that there must be something global that influences the path of particles, such that a single particle going through one of the slits seems to take into account whether the other slit is covered or not. But before you can have a particle trajectory you must have particles. Virtual particles seem to pop into and out of existence as part of the zero point energy. They can be made real particles if one of the pair is captured by a horizon. So it seems we are looking for some global mechanism for particle creation in the first place. And the same global topological concerns that give rise to virtual particles to begin with should incorporate some dynamics to account for trajectories of real particles to end with. So I consider what topological entities might give rise to particles. I think in terms of an index theorm or some AdS/CFT effect going on.

When I think of the first particles arising from the tiny, expanding universe, it seems that whatever the mechanism of virtual particle creation, it must some how proceed in a smooth way from a singularity. The first fluctuations would be in the size and location of the entire, tiny spacetime of the universe. There would not be room enough, yet for virtual particles, and the fluxuation would simply be in some degree of freedom in the boundary or overall size of the tiny universe. Perhaps this is the same mechanism that forces the universe to expand. Then as the universe becomes large enough, these fluctuations can include particles that pop in and out of existence as these overall topological entities change.

 

[marcus]

Actually, it seems that Torsten has only recently appended the second part of his surname.

Good. so he has some halfdozen papers on arxiv, and they go back to 1995.

Helge said that Torsten was writing a book.
Exotic Structures and Physics: Differential Topology and Spacetime Models
I can make better sense of that now that I know he has been working in this general field for 10 years.

 

[Helge Rosé]

Thats right, we think about the topic nearly 15 years. Torstens DS idea is younger (10 years). Since 2 years he could show that the DS should build a Hilbert-space. The Temperley-Lieb-algebra structure of the changes is from this year.

 

[marcus]

Thats right, we think about the topic nearly 15 years. Torstens DS idea is younger (10 years). Since 2 years he could show that the DS should build a Hilbert-space. The Temperley-Lieb-algebra structure of the changes is from this year.

it is after midnight and i have to sleep
I hope you return here tomorrow. I will try to have some questions.
I don't understand how a 3D submanif. can represent a particle
but no use explaining now, I will give it another try tomorrow

 

[Helge Rosé]

This (post #6) is correct. We are work at FIRST in Berlin. We have published papers about different topics (Evolutionary Algorithms, Quantum-Hall-effect, Quantum Computing, Computer-stuff like simulation of complex systems ...) e.g. http://www.first.fhg.de/helge.rose/publications
or my home-page http://www.first.fhg.de/helge.rose
We have not published our ideas to QG (except Torstens 1996 paper) until now because - well it is not an easy topic and very explosive. We would like to make the ideas save to form a whole picture. The last steps (Temperly-Lieb-algebra of DS transitions) are appeared this year. This paper is only a first step - so to say the "kinematik" of the theory. We have ideas to the next steps - the dynamics, i.e. the field equation. Only this will complete the picture and hopefully get a new usefull theory. But we think the results from the 0506067-paper are promissing. I think the interessting discussion here will be very fruitfull for this.

 

[Helge Rosé]

I don't understand how a 3D submanif. can represent a particle

this is non trivial. Could you explain a little more your question.

 

[marcus]

this is non trivial. Could you explain a little more your question.

it is too early to ask that question and I should back up a little and pick something earlier, like on page 4, beginning of section III.

"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.

 

[Helge Rosé]

Do you have any suggestions of papers to read as preparation for your paper with Torsten?

So far I just see the citations to papers by Brans and by Sladowski (including one that Torsten co-authored with Brans)
are these the best to read or are there also others you might suggest?

The work is distibuted over the math. literature. Torsten and Carl Brans wrote the book to give a review about this field. I think torsten should collect some papers, I will ask him.

I will try to answer your post about 3d-support next morning (at night of your time).

 

[garrett]

This discussion started on another thread, but I thought it best to bring it over here:

Hi garrett, thanks for your interest in our paper.
In the physical point of view an atlas is set of reference frames which are needed to describe measurements at different space-time regions. With Einstein all reference frames are physically equal if the charts can be transformed by diffeomorphisms - the charts are compatible.
In 1,2,3 dimensions all charts (reference frames) are compatible. In 4 dimensions you can find one set S1 of charts which are compatible with all other charts in this set. But you can also find a further set S2, were all charts compatible in S2 but with no chart in S1. S1 and S2 are two different representants of two atlases. S1 and S2 belong to two different differential structures.
In mathematics the differential structures are called exotic smooth strctures. It can be shown that for a manifold (e.g. Dim=7 like Milnor) atlases exist which are not compatible (transfromable by diffeomorphisms). It is also known that for a compact 4-manifold the number of non-compatible altlases are countable infinite. But the structure of the set of differential structures was unknown. We have shown that the set of the changes of a differential structure is a Temperley-Lieb algebra and the set of differential structures is a Hilbert-space (Dim H = inf). This is a mathematical fact like: "the number of integers is countable infinite". Torsten is writing a book about https://www.amazon.com/gp/product/981024195X/?tag=pfamazon01-20
At the mean time you may looking for the mathematical papers about "Exotic Structures", but this is hard to cover.

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:

As an important fact we will note that there is only one differential structure of any manifold of dimension smaller than four. For all manifolds larger than four dimensions there is only a finite number of possible differential structures, Diff_{dim M}. The following table lists the numbers of differential structures up to dimension 11.
1 1 1 inf 1 1 28 2 8 6 992

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"
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.

 

[Kea]

it may take days before my brain stops smoking and making sparks and begins to understand...

Only days? Marcus, you're a wizard!

 

[Careful]

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.

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. 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. 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...

 

[Kea]

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...I think these issues need clarification otherwise it seems to go wrong from the beginning...

Careful

I think it is only fair to judge this paper by physicists' standards, and by these standards it is extremely interesting. Moreover, Helge and Torsten do not claim to have provided the motivation, the depth of which would take volumes.

Anyway, all good String theorists are working with twistors these days, no?

 

[Kea]

People may be interested in an old thread on the spin foam connection

http://www.lns.cornell.edu/spr/2003-10/msg0055272.html

 

[Careful]

Careful

I think it is only fair to judge this paper by physicists' standards, and by these standards it is extremely interesting. Moreover, Helge and Torsten do not claim to have provided the motivation, the depth of which would take volumes.

Anyway, all good String theorists are working with twistors these days, no?

Kea, you do not seem to have understood that I am questioning the PHYSICAL motivation for doing this (actually, I have a whole other page of objections). Moreover, as far as I see, the BASIC mathematical ideas in their theory are wrong (I would be happy to receive detailed comments). Hence, unless these issues are resolved, there is no theory and therefore no physical relevance.

There is something to say however, for the idea of matter being represented by objects with a singularity; but this has not necessarily anything to do with a change of differentiable structure.

PS: I am not a string theorist, but I am a physicist.

 

[Kea]

Kea, you do not seem to have understood that I am questioning the PHYSICAL motivation for doing this.

Well, hopefully Helge will return shortly and we can begin discussing this.

 

[marcus]

Yes, I am excited about these new deep results. For example Asselmeyer and Rose' show that their approach restricts the underlying coefficient module to be the complex numbers, ...

can you (or some volunteer) please pedagogically explain how they manage to show the compex numbers are the inevitable coefficients?
I read where they assert this but would appreciate a little help with how they get there.

 

[Kea]

A quote from one of the Brans papers (reference 3)

"What is important for our discussion is that this standard model [of $R^4$] implicitly imparts to each spacetime point an identifiable, objective existence, independent of any choice of coordination. In fact, this topological notion is prior to the existence of any coordinates...A thorough understanding of this fact is absolutely necessary to follow the intracacies involved with the definition of differentiable structures."

 

[Kea]

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.

 

[Careful]

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.

No, not at all ! But why do you not give the honor to Helge to answer my questions? I think my mathematical objections are serious enough.

 

[Helge Rosé]

No, not at all ! 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.

 

[Helge Rosé]

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.

 

[Helge Rosé]

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"
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.

 

[Helge Rosé]

"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 - that's your
"FAULT" of earthquakes. But don't 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! I am sure torsten will explain that again and much better.

 

[selfAdjoint]

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 - that's your
"FAULT" of earthquakes. But don't 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! I am 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"

And http://mathworld.wolfram.com/h-CobordismTheorem.html" , 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"

 

[garrett]

Hi Helge, thanks for addressing these issues.

The wikipedia entry on differential structures is now fixed:
http://en.wikipedia.org/wiki/Differential_structure

 

[marcus]

...
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 *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

 

[selfAdjoint]

See why the Poincare conjecture in dim 4 is important?