この中に www.nilesjohnson.net/seven-manifolds.html An animation of exotic 7-spheres Video from a presenation by Niles Johnson at the Second Abel conference in honor of John Milnor.
mas.lvc.edu/~lyons/pubs/ David W. Lyons' Publications and Presentations Selected Expository Work [1] David W. Lyons. An elementary introduction to the Hopf fibration. Mathematics Magazine, 76(2):87-98, 2003. [ journal | e-print ]
In mathematics, Seiberg?Witten invariants are invariants of compact smooth 4-manifolds introduced by Witten (1994), using the Seiberg?Witten theory studied by Seiberg and Witten (1994a, 1994b) during their investigations of Seiberg?Witten gauge theory.
Seiberg?Witten invariants are similar to Donaldson invariants and can be used to prove similar (but sometimes slightly stronger) results about smooth 4-manifolds. They are technically much easier to work with than Donaldson invariants; for example, the moduli spaces of solutions of the Seiberg?Witten equations tend to be compact, so one avoids the hard problems involved in compactifying the moduli spaces in Donaldson theory.
For detailed descriptions of Seiberg?Witten invariants see (Donaldson 1996), (Moore 2001), (Morgan 1996), (Nicolaescu 2000), (Scorpan 2005, Chapter 10). For the relation to symplectic manifolds and Gromov?Witten invariants see (Taubes 2000). For the early history see (Jackson 1995).
This is the home page for the Polymath8 project, which has two components:
Polymath8a, "Bounded gaps between primes", was a project to improve the bound H=H_1 on the least gap between consecutive primes that was attained infinitely often, by developing the techniques of Zhang. This project concluded with a bound of H = 4,680.
Polymath8b, "Bounded intervals with many primes", is an ongoing project to improve the value of H_1 further, as well as H_m (the least gap between primes with m-1 primes between them that is attained infinitely often), by combining the Polymath8a results with the techniques of Maynard.
Contents 1 World records 1.1 Current records 1.2 Timeline of bounds 2 Polymath threads 3 Writeup 4 Code and data 4.1 Tuples applet 5 Errata 6 Other relevant blog posts 7 MathOverflow 8 Wikipedia and other references 9 Recent papers and notes 10 Media 11 Bibliography
つづき levitopher.wordpress.com/2013/05/08/exotic-smoothness-iii-existence/ May 8, 2013 by cduston Exotic Smoothness III: Existence Dimension 4 So the problem is that decomposition techniques generally fail in dimension 4, due to the added complexity but failure of the Whitney disk trick. Now, the topological version of the h-cobordism theorem works; meaning that two manifolds that are homotopic in dimension four are also homeomorphic. Of course, that doesn’t help us very much because we are at least in the category of continuity; want we want is the difference between continuous and smooth. 略 By a complete classification of these forms (done by Freedman and Donaldson), you can do things like try and decompose the manifold while preserving the intersection form. This leads to some contradictions, the most interesting of which leads to the existence of exotic R^4. These would be smooth 4-manifolds which are homeomorphic to our usual R^4, but which are not diffeomorphic to the usual R^4. Things are even worse (or better!) ? there are infinitely-many exotic R^4!
So the situation is this; in terms of exotic smoothness, dimension 4 is special. This presents a major motivation for studying exotic smoothness in the context of physics. We have already discussed that since exotic smooth structures are not smoothly equivalent, we would not expect any results which relied on calculus (like physics!) to be the same on both of them. Of course, this would not matter if we were studying the physics of space alone ? since it is 3-dimensional, there is no exotic smoothness. But as soon as we move to the dimension in which all our fundamental theories are based, exotic smoothness suddenly becomes non-trivial.
This is either a very significant observation, or it is not! The next post will discuss how we might try to study exotic smoothness in physics, from both model-building and observational standpoints.
So far, I have introduced some of the basic notions of smooth manifolds, what exotic smoothness is, and (very superficially!) how we know it exists. In this post I will talk about how one can go about constructing a physical model which includes exotic smooth structures, and what kinds of behavior we can expect. “What problems can exotic smoothness solve?” might be a summary for this post, but as we will see, there is more conjecture then problem solving.
Large and Small Exotic R^4 略
Dark Matter 略
The Brans Conjecture Localized exotic smoothness can mimic an additional source for the gravitational field. Of course, this conjecture is quite vague, but what Brans had in mind was exactly a solution to the dark matter problem. 略
Normal Matter
I think it’s fair to say the Brans conjecture has not been proven yet ? specifically, there is not currently a model of dark matter which can be compared to (and thus verified by) observations. However, there has certainly been work done which suggests that exotic smoothness can mimic mass in more limited ways. For instance, Torsten Asselmeyer-Maluga (you will see his name come up frequently in connection with this topic ? he has been diligently working on getting very interesting results for over a decade now) has shown that the intersection of some special surfaces in 4-manifolds (which represent points of which a homeomorphism f:M →M' fails to be a diffeomorphism) can create non-zero curvature terms (1997). In other words, the failure of two 4-manifolds to be diffeomorphic at points can mimic mass terms. This can be extended (see here and here), so that it appears that this result is quite general, and can be used to construct matter with a variety of internal symmetries.
Thus, in semiclassical gravity there are at least some instances when the Bran Conjecture is certainly true.
Well, this was long post but I wanted to give the current state of model-building based on exotic smooth structures. I think I will stop here; much of my other work is related to this topic, but this is enough to know in terms of exotic smooth structure.
有名なJohn Milnorが、”The Poincare Conjecture 99 Years Later”を書いて、そのときPerelman の論文が投稿されて、半信半疑だと書いていた・・ www.math.sunysb.edu/~jack/PREPRINTS/poiproof.pdf The Poincare Conjecture 99 Years Later: A Progress Report John Milnor, Stony Brook University, February 2003
抜粋 Three months ago, Grisha Perelman in St. Petersburg posted a preprint describing a way to resolve some of the major stumbling blocks in the Hamilton program and suggesting a path toward a solution of the full Elliptization Conjecture. The initial response of experts to this claim has been carefully guarded optimism, although, in view of the long history of false proofs in this area, no one will be convinced until all of the details have been carefully explained and veri￣ed. Perelman is planning to visit the United States in April, at which time his arguments will no doubt be subjected to detailed scrutiny.
ユーザープログラムを配信・利用できるCRANネットワーク機能 世界中のRユーザが開発したRプログラム（ライブラリ）（これを「パッケージ」と呼ぶ）がCRAN (The Comprehensive R Archive Network) と呼ばれるネットワークで配信されており、 それらをR環境単独でオンラインでダウンロード・インストール・アップグレードと一連の管理が可能である。
４次元で、下記が、なかなか絵が充実しているね これが置いてある、Andrew Ranicki’s Homepageでも”wild ”という言葉が良くヒットする。”wild ”すきみたいだね。すぎちゃんの系統かね www.maths.ed.ac.uk/~aar/papers/scorpan.pdf Scorpan, A. (2005), The wild world of 4-manifolds, Providence, R.I.: American Mathematical Society, ISBN 0-8218-3749-4
the following are rough draft versions of a text-to-be on applied algebraic topology, all in pdf. enjoy! the bibliographic entries are not yet added, and some of the cross-references and pictures are muddled...sorry!
下記がよくまとまっている en.wikipedia.org/wiki/4-manifold#cite_note-2 （抜粋） 4-manifold From Wikipedia, the free encyclopedia Contents 1 Topological 4-manifolds 2 Smooth 4-manifolds 3 Special phenomena in 4-dimensions 4 Failure of the Whitney trick in dimension 4
Smooth 4-manifolds For manifolds of dimension at most 6, any piecewise linear (PL) structure can be smoothed in an essentially unique way,[1] so in particular the theory of 4 dimensional PL manifolds is much the same as the theory of 4 dimensional smooth manifolds. A major open problem in the theory of smooth 4-manifolds is to classify the simply connected compact ones. As the topological ones are known, this breaks up into two parts:
1. Which topological manifolds are smoothable? 2. Classify the different smooth structures on a smoothable manifold.
There is an almost complete answer to the first problem of which simply connected compact 4-manifolds have smooth structures.
>>535 > 1. Which topological manifolds are smoothable? >There is an almost complete answer to the first problem of which simply connected compact 4-manifolds have smooth structures.
>>541 ども 4次元だとキャッソンハンドルに同相らしい "Casson's idea was to iterate this construction an infinite number of times, in the hope that the problems about double points will somehow disappear in the infinite limit." ということなので、”disappear in the infinite limit."が結論だな
en.wikipedia.org/wiki/Casson_handle In 4-dimensional topology, a branch of mathematics, a Casson handle is a 4-dimensional topological 2-handle constructed by an infinite procedure. They are named for Andrew Casson, who introduced them in about 1973. They were originally called "flexible handles" by Casson himself, and Michael Freedman (1982) introduced the name "Casson handle" by which they are known today. In that work he showed that Casson handles are topological 2-handles, and used this to classify simply connected compact topological 4-manifolds.
en.wikipedia.org/wiki/N-sphere#Spherical_coordinates An n-sphere is the surface or boundary of an (n + 1)-dimensional ball, and is an n-dimensional manifold. For n ≥ 2, the n-spheres are the simply connected n-dimensional manifolds of constant, positive curvature. The n-spheres admit several other topological descriptions: for example, they can be constructed by gluing two n-dimensional Euclidean spaces together, by identifying the boundary of an n-cube with a point, or (inductively) by forming the suspension of an (n − 1)-sphere.
4-sphere Equivalent to the quaternionic projective line, HP1. SO(5)/SO(4).
en.wikipedia.org/wiki/Quaternionic_projective_line In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H. Quaternionic projective space of dimension n is usually denoted by
\mathbb{HP}^n
and is a closed manifold of (real) dimension 4n. It is a homogeneous space for a Lie group action, in more than one way.
Projective line From the topological point of view the quaternionic projective line is the 4-sphere, and in fact these are diffeomorphic manifolds. The fibration mentioned previously is from the 7-sphere, and is an example of a Hopf fibration.
As mentioned above, in dimensions smaller than 4, there is only one differential structure for each topological manifold. That was proved by Johann Radon for dimension 1 and 2, and by Edwin E. Moise in dimension 3.[3] By using obstruction theory, Robion Kirby and Laurent Siebenmann [4] were able to show that the number of PL structures for compact topological manifolds of dimension greater than 4 is finite. John Milnor, Michel Kervaire, and Morris Hirsch proved that the number of smooth structures on a compact PL manifold is finite and agrees with the number of differential structures on the sphere for the same dimension (see the book Asselmeyer-Maluga, Brans chapter 7) By combining these results, the number of smooth structures on a compact topological manifold of dimension not equal to 4 is finite.
Dimension 4 is more complicated. For compact manifolds, results depend on the complexity of the manifold as measured by the second Betti number b_2. For large Betti numbers b_2>18 in a simply connected 4-manifold, one can use a surgery along a knot or link to produce a new differential structure. With the help of this procedure one can produce countably infinite many differential structures. But even for simple spaces like S^4, {\mathbb C}P^2,... one doesn't know the construction of other differential structures. For non-compact 4-manifolds there are many examples like {\mathbb R}^4,S^3\times {\mathbb R},M^4\setminus\{*\},... having uncountably many differential structures.
Chapter 11 From Differential Structures to Operator Algebras and Geometric Structures This chapter surveys some of the interesting interplay of exotic smoothness with other areas of mathematics and physics. In the first section we consider the “change” of a differential structure on a given TOP manifold to a differential structure on a second manifold homeomorphic but not diffeomorphic to the first one. Harvey and Lawson introduced the notion of singular bundle maps and connections to study this problem. This leads to speculations that such a process could give rise to singular string-like sources to the Einstein equations of General Relativity, including torsion. The next section deals with formal properties of a connection change and its relation to cyclic cohomology, providing a relationship between Casson handles and Ocneanus string algebra. This approach motivates introduction of the hyperfinite II1 factor C* algebra T leading to the conjecture that the differential structures are classified by the homotopy classes [M, BGl(T)+]. This conjecture may have some significance for the the 4-dimensional, smooth Poincark conjecture. The last section introduces a conjecture relating differential structures on 4-manifolds and geometric structures of homology 3-spheres naturally embedded in them.
About The differential or smoothness structure of a topological manifold (if it exists) can be non-unique. In all dimension except 4 there are only a finite number of different (i.e. non-diffeomorphic) smoothness structures. But dimension 4 is exceptional. Here there are an infinite number of different smoothness structures, countable infinite for most compact and uncountable many for many non-compact 4-manifolds. But what is the physical meaning of this fact, that is my main research program.
en.wikipedia.org/wiki/Carl_H._Brans Carl Henry Brans (born December 13, 1935) is an American mathematical physicist best known for his research into the theoretical underpinnings of gravitation elucidated in his most widely publicized work, the Brans–Dicke theory.
Recently Brans began study of developments in differential topology concerning the existence of exotic (non-standard) global differential structures and their possible applications to physics. This work includes looking at the exotic 7-sphere of Milnor as an exotic Yang-Mills bundle, and most especially the infinity of exotic differential structure on Euclidean four space (exotic R4) as alternative models for space-time in general relativity. Much of this work has been done in collaboration with Torsten Asselmeyer-Maluga of Berlin. In particular, they made the proposal that exotic smoothness structures can be resolve some of the problems in cosmology like dark matter or dark energy. Together they published a book, Exotic Smoothness and Physics World Scientific Press, 2007.
