Research Article Inflation and Topological Phase...

Research ArticleInflation and Topological Phase TransitionDriven by Exotic Smoothness

Torsten Asselmeyer-Maluga1 and Jerzy Kroacutel2

1 German Aerospace Center (DLR) 10178 Berlin Germany2 Institute of Physics University of Silesia 40-007 Katowice Poland

Correspondence should be addressed to Torsten Asselmeyer-Maluga torstenasselmeyer-malugadlrde

Received 5 December 2013 Accepted 17 January 2014 Published 19 March 2014

Academic Editor Douglas Singleton

Copyright copy 2014 T Asselmeyer-Maluga and J Krol This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited The publication of this article was funded by SCOAP3

We will discuss a model which describes the cause of inflation by a topological transition The guiding principle is the choice of anexotic smoothness structure for the space-time Here we consider a space-time with topology 1198783 times R In case of an exotic 1198783 times Rthere is a change in the spatial topology from a 3-sphere to a homology 3-sphere which can carry a hyperbolic structure Fromthe physical point of view we will discuss the path integral for the Einstein-Hilbert action with respect to a decomposition of thespace-timeThe inclusion of the boundary terms produces fermionic contributions to the partition functionThe expectation valueof an area (with respect to some surface) shows an exponential increase that is we obtain inflationary behavior We will calculatethe amount of this increase to be a topological invariantThen we will describe this transition by an effective model the Starobinskior 1198772 model which is consistent with the current measurement of the Planck satellite The spectral index and other observables arealso calculated

1 Introduction

General relativity (GR) has changed our understanding ofspace-time In parallel the appearance of quantum fieldtheory (QFT) has modified our view of particles fieldsand the measurement process The usual approach for theunification of QFT and GR to a quantum gravity startswith a proposal to quantize GR and its underlying structurespace-time There is a unique opinion in the communityabout the relation between geometry and quantum theorythe geometry as used in GR is classical and should emergefrom a quantum gravity in the limit (Planckrsquos constant tendsto zero) Most theories went a step further and try to geta space-time from quantum theory Then the model of asmooth manifold is not suitable to describe quantum gravityBut there is no sign for a discrete space-time structure orhigher dimensions in current experiments Hence quantumgravity based on the concept of a smooth manifold shouldalso able to explain the current problems in the standardcosmological model (ΛCDM) like the appearance of darkenergymatter or the correct form of inflation But before we

are going in this direction we will motivate the usage of thesmooth manifold as our basic concept

When Einstein developed GR his opinion about theimportance of general covariance changed over the yearsIn 1914 he wrote a joint paper with Grossmann There herejected general covariance by the now famous hole argu-ment But after a painful year he again considered generalcovariance now with the insight that there is no meaning inreferring to the space-time point A or the event A withoutfurther specifications Therefore the measurement of a pointwithout a detailed specification of the whole measurementprocess is meaningless in GR The reason is simply thediffeomorphism invariance of GR which has tremendousconsequences Furthermore GR do not depend on thetopology of space-time All restrictions on the topology ofthe space-time were formulated using additional physicalconditions like causality (see [1]) This ambiguity increasesin the 80rsquos when the first examples of exotic smoothnessstructures in dimension 4 were found The (smooth) atlas ofa smooth 4-manifold 119872 is called the smoothness structure(unique up to diffeomorphisms)Onewould expect that there

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 867460 14 pageshttpdxdoiorg1011552014867460

2 Advances in High Energy Physics

is only one smooth atlas for 119872 all other possibilities canbe transformed into each other by a diffeomorphism But incontrast the deep results of Freedman [2] on the topologyof 4-manifolds combined with Donaldsonrsquos work [3] gave thefirst examples of nondiffeomorphic smoothness structureson 4-manifolds including the well-known R4 Much of themotivation can be found in the FQXI essay [4 5] Herewewilldiscuss another property of the exotic smoothness structureits quantum geometry in the path integral

Diffeomorphism invariance is the most important prop-erty of the Einstein-Hilbert action with far reaching conse-quences [6] One of our results is a close relation betweengeometry and foliation to exotic smoothness [7 8] In theparticular example of the exotic R4 we discussed the exoticsmoothness structures as a manifestation of quantum gravity(by using string theory [9 10]) This exotic R4 has someinteresting properties as first noted by Brans [11 12] Moreimportantly as shown by Sładkowski [13] the exotic R4 hasa nontrivial curvature in contrast to the flat standard R4It was the first result that an exotic R4 can be seen as asource of gravity (or it must contain sources of gravity)Sładkowski [14ndash16] went further and showed a relation toparticle physics also related to quantum gravity But whythere is a relation to quantum gravity In [17] we presentedthe first idea to understand this relation which was furtherextended in [18] An exotic 4-manifold like 1198783 times R is alsocharacterized by the property that there is no smoothlyembedded 3-sphere but a topological embedded one Thistopological 1198783 is wildly embedded that is the image of theembedding must be triangulated by an infinite polyhedronIn [18] we proved that the (deformation) quantization of ausual (or tame) embedding is a wild embedding which canbe seen as a quantum state But then any exotic 4-manifoldcan be interpreted as a quantum state of the 4-manifoldwith standard smoothness structure From this point ofview the calculation of the path integral in quantum gravityhas to include the exotic smoothness structures Usually itis hopeless to make these calculations But by using theclose relation of exotic smoothness to hyperbolic geometryone has a chance to calculate geometric expressions likethe expectation value of the surface area In this paper wewill show that this expectation value has an inflationarybehavior that is the area grows exponentially (along thetime axis) Therefore quantum gravity (in the sense of exoticsmoothness) can be the root of inflation

2 Space-Time and Smoothness

From the mathematical point of view the space-time isa smooth 4-manifold endowed with a (smooth) metric asbasic variable for general relativity The existence questionfor Lorentz structure and causality problems (see Hawkingand Ellis [1]) give further restrictions on the 4-manifoldcausality implies noncompactness Lorentz structure needsa nonvanishing normal vector field Both concepts can becombined in the concept of a global hyperbolic 4-manifold119872 having a Cauchy surface S so that119872 = S timesR

All these restrictions on the representation of space-timeby the manifold concept are clearly motivated by physicalquestions Among these properties there is one distinguishedelement the smoothness Usually one starts with a topolog-ical 4-manifold119872 and introduces structures on them Thenone has the following ladder of possible structures

Topology 997888rarr piecewise-linear (PL) 997888rarr Smoothness

997888rarr bundles Lorentz Spin etc

997888rarr metric geometry

(1)

We do not want to discuss the first transition that is theexistence of a triangulation on a topological manifold But weremark that the existence of a PL structure implies uniquelya smoothness structure in all dimensions smaller than 7 [19]Herewe have to consider the following steps to define a space-time

(i) Fix a topology for the space-time119872(ii) Fix a smoothness structure that is a maximal differ-

entiable atlasA(iii) Fix a smoothmetric or get one by solving the Einstein

equation

The choice of a topology never fixes the space-time uniquelythat is there are two space-times with the same topologywhich are not diffeomorphicThemain idea of the paper is theintroduction of exotic smoothness structures into space-timeIf two manifolds are homeomorphic but nondiffeomorphicthey are exotic to each other The smoothness structure iscalled an exotic smoothness structure

In dimension four there are many examples of com-pact 4-manifolds with countable infinite nondiffeomorphicsmoothness structures andmany examples of noncompact 4-manifolds with uncountable infinite many nondiffeomorphicsmoothness structures But in contrast the number of non-diffeomorphic smoothness structures is finite for any otherdimension [19] As an example we will consider the space-time 1198783 times R having uncountable many nondiffeomorphicsmoothness structures in the following

3 The Path Integral in Exotic 1198783 timesR

For simplicity we consider general relativity without matter(using the notation of topological QFT) Space-time is asmooth oriented 4-manifold 119872 which is noncompact andwithout boundary From the formal point of view (no diver-gences of the metric) one is able to define a boundary 120597119872 atinfinity The classical theory is the study of the existence anduniqueness of (smooth)metric tensors119892 on119872 that satisfy theEinstein equations subject to suitable boundary conditionsIn the first order Hilbert-Palatini formulation one specifiesan SO(1 3)-connection 119860 together with a cotetrad field 119890rather than a metric tensor Fixing 119860|

120597119872at the boundary

one can derive first-order field equations in the interior (nowcalled bulk) which are equivalent to the Einstein equationsprovided that the cotetrad is nondegenerate The theory isinvariant under space-time diffeomorphisms 119872 rarr 119872

Advances in High Energy Physics 3

In the particular case of the space-time 119872 = 1198783times R

(topologically) we have to consider a smooth 4-manifolds119872

119894119891as parts of119872 whose boundary 120597119872

119894119891= Σ

119894⊔ Σ

119891is the

disjoint union of two smooth 3-manifolds Σ119894and Σ

119891to which

we associate Hilbert spaces H119895of 3-geometries 119895 = 119894 119891

These contain suitable wave functionals of connections 119860|Σ119895

We denote the connection eigenstates by |119860|Σ119895⟩ The path

integral

⟨119860|Σ119891

10038161003816100381610038161198791198721003816100381610038161003816 119860|Σ119894

⟩

= int119860|120597119872119894119891

119863119860119863119890 exp( 119894ℎ119878119864119867[119890 119860119872

119894119891])

(2)

is the sum over all connections 119860 matching 119860|120597119872119894119891

andover all 119890 It yields the matrix elements of a linear map119879119872 H

119894rarr H

119891between states of 3-geometry Our basic

gravitational variables will be cotetrad 119890119868119886and connection119860119868119869

119886

on space-time 119872 with the index 119886 to present it as 1-formsand the indices 119868 119869 for an internal vector space 119881 (used forthe representation of the symmetry group) Cotetrads 119890 areldquosquare-rootsrdquo of metrics and the transition from metrics totetrads is motivated by the fact that tetrads are essential ifone is to introduce spinorial matter 119890119868

119886is an isomorphism

between the tangent space 119879119901(119872) at any point 119901 and a fixed

internal vector space 119881 equipped with a metric 120578119868119869so that

119892119886119887= 119890

119868

119886119890119869

119887120578119868119869 Here we used the action

119878119864119867[119890 119860119872

119894119891 120597119872

119894119891]

= int119872119894119891

120598119868119869KL (119890

119868and 119890

119869and (119889119860 + 119860 and 119860)

KL)

+ int120597119872119894119891

120598119868119869KL (119890

119868and 119890

119869and 119860

KL)

(3)

in the notation of [20 21] Here the boundary term 120598119868119869KL(119890

119868and

119890119869and119860

KL) is equal to twice the trace over the extrinsic curvature

(or the mean curvature) For fixed boundary data (2) isa diffeomorphism invariant in the bulk If Σ

119894= Σ

119891are

diffeomorphic we can identify Σ = Σ119894= Σ

119891and H = H

119894=

H119891 that is we close themanifold119872

119894119891by identifying the two

boundaries to get the closed 4-manifold1198721015840 Provided that thetrace overH can be defined the partition function

119885(1198721015840) = trH119879119872 = int119863119860119863119890 exp( 119894

ℎ119878119864119867 [119890 119860119872 120597119872])

(4)

where the integral is now unrestricted is a dimensionlessnumber which depends only on the diffeomorphism class ofthe smooth manifold 1198721015840 In case of the manifold 119872

119894119891 the

path integral (as transition amplitude) ⟨119860|Σ119891|119879

119872|119860|

Σ119894⟩ is the

diffeomorphism class of the smooth manifold relative to theboundary But the diffeomorphism class of the boundary isunique and the value of the path integral depends on the

topology of the boundary as well on the diffeomorphism classof the interior of119872

119894119891 Therefore we will shortly write

⟨Σ119891

10038161003816100381610038161198791198721003816100381610038161003816 Σ119894⟩ = ⟨119860|Σ119891

10038161003816100381610038161198791198721003816100381610038161003816 119860|Σ119894

⟩ (5)

and consider the sum of manifolds like119872119894ℎ= 119872

119894119891cupΣ119891119872

119891ℎ

with the amplitudes

⟨Σℎ

10038161003816100381610038161198791198721003816100381610038161003816 Σ119894⟩ = sum

119860|Σ119891

⟨Σℎ

10038161003816100381610038161198791198721003816100381610038161003816 Σ119891⟩ ⟨Σ119891

10038161003816100381610038161198791198721003816100381610038161003816 Σ119894⟩ (6)

where we sum (or integrate) over the connections and frameson Σ

ℎ(see [22]) Then the boundary term

119878120597[Σ

119891] = int

Σ119891

120598119868119869KL (119890

119868and 119890

119869and 119860

KL) = int

Σ119891

119867radicℎ1198893119909 (7)

is neededwhere119867 is themean curvature ofΣ119891corresponding

to themetric ℎ atΣ119891(as restriction of the 4-metric)Therefore

we have to divide the path integration into two parts thecontribution by the boundary (boundary integration) and thecontribution by the interior (bulk integration)

31 Boundary Integration The boundary Σ of a 4-manifold119872 can be understood as embedding (or at least as immer-sion) Let 120580 Σ 997893rarr 119872 be an immersion of the 3-manifoldΣ into the 4-manifold 119872 with the normal vector

rarr

119873 Thespin bundle 119878

119872of the 4-manifold splits into two subbundles

119878plusmn

119872 where one subbundle say 119878+

119872 can be related to the spin

bundle 119878Σof the 3-manifoldThen the spin bundles are related

by 119878Σ= 120580

lowast119878+

119872with the same relation 120601 = 120580

lowastΦ for the

spinors (120601 isin Γ(119878Σ) and Φ isin Γ(119878

+

119872)) Let nabla119872

119883 nabla

Σ

119883be the

covariant derivatives in the spin bundles along a vector field119883 as section of the bundle 119879Σ Then we have the formula

nabla119872

119883(Φ) = nabla

Σ

119883120601 minus

1

2(nabla

119883

rarr

119873) sdotrarr

119873 sdot120601 (8)

with the obvious embedding 120601 997891rarr (120601

0) = Φ of the spinor

spacesThe expressionnabla119883

rarr

119873 is the second fundamental formof the immersion where the trace tr(nabla

119883

rarr

119873) = 2119867 is relatedto themean curvature119867Then from (8) one obtains a similarrelation between the corresponding Dirac operators

119863119872Φ = 119863

3119863120601 minus 119867120601 (9)

with the Dirac operator 1198633119863 of the 3-manifold Σ Near theboundary Σ the 4-manifolds looks like Σtimes[0 1] and a spinorΦ on this 4-manifold is a parallel spinor and has to fulfill thefollowing equation

119863119872Φ = 0 (10)

that is 120601 yields the eigenvalue equation

1198633119863120601 = 119867120601 (11)

with the mean curvature119867 of the embedding 120580 as eigenvalueSee our previous work [23] for more details


Now we will use this theory to get rid of the boundaryintegration At first we will discuss the deformation of animmersion using a diffeomorphism Let 119868 Σ 997893rarr 119872

be an immersion of Σ (3-manifold) into 119872 (4-manifold)A deformation of an immersion 119868

1015840 Σ

1015840997893rarr 119872

1015840 isdiffeomorphisms 119891 119872 rarr 119872

1015840 and 119892 Σ rarr Σ1015840 of119872 and Σ

respectively so that

119891 ∘ 119868 = 1198681015840∘ 119892 (12)

One of the diffeomorphisms (say 119891) can be absorbedinto the definition of the immersion and we are left withone diffeomorphism 119892 isin Diff(Σ) to define the deformationof the immersion 119868 But as stated above the immersion isdirectly given by an integral over the spinor 120601 on Σ fulfillingthe Dirac equation (11) Therefore we have to discuss theaction of the diffeomorphism group Diff(Σ) on the Hilbertspace of 1198712-spinors fulfilling the Dirac equation This casewas considered in the literature [24]The spinor space 119878

119892120590(Σ)

on Σ depends on two ingredients a (Riemannian) metric 119892and a spin structure 120590 (labeled by the number of elementsin 119867

1(ΣZ

2)) Let us consider the group of orientation

preserving diffeomorphismDiff+(Σ) acting on 119892 (by pullback119891lowast119892) and on 120590 (by a suitable defined pullback 119891lowast

120590) TheHilbert space of 1198712-spinors of 119878

119892120590(Σ) is denoted by 119867

119892120590

Then according to [24] any 119891 isin Diff+(Σ) leads in exactlytwo ways to a unitary operator 119880 from 119867

119892120590to 119867

119891lowast119892119891lowast120590

The (canonically) defined Dirac operator is equivariant withrespect to the action of119880 and the spectrum is invariant under(orientation preserving) diffeomorphisms In particular weobtain for the boundary term

119878120597[Σ

119891 ℎ] = int

Σ119891

119867radicℎ1198893119909 = int

Σ119891

120601 1198633119863120601119889

3119909 (13)

with |120601|2 = const (see [25]) But then we can change theintegration process from the integration over the metric classℎ on the 3-manifoldΣ

119891withmean curvature to an integration

over the spinor 120601 on Σ119891 Then we obtain

119885(Σ119891) = int119863ℎ exp( 119894

ℎ119878 [Σ

119891 ℎ])

= int119863120601119863120601 exp( 119894ℎintΣ119891

120601 1198633119863120601119889

3119909)

= radicdet (1198633119863119863lowast3119863)119890119894120587120578(Σ119891)2

(14)

where 120578(Σ119891) is the Eta invariant of the Dirac operator at the

3-manifold Σ119891(here we use a result of Witten see [26])

From the physical point of view we obtain fermions at theboundary The additional term with the Eta invariant reflectsalso an important fact The state space of general relativity isthe space of the (Lorentzian) metric tensor up to the groupof coordinate transformations This group of coordinatetransformations is not the full diffeomorphism group it isonly one connected component of the diffeomorphism Thatis the group of diffeomorphisms connected to the identityIn addition there is also the (discrete) group of global

diffeomorphisms which is in our case detected by the Etainvariant For 3-manifolds there is a deep relation to theChern-Simons invariant [27] which will be further studied atour forthcoming work

32 Bulk Integration Nowwewill discuss the path integral ofthe action

119878119864119867 [119890 119860119872] = int

119872

120598119868119869KL (119890

119868and 119890

119869and (119889119860 + 119860 and 119860)

KL) (15)

in the interior of the 4-manifold119872 The contribution of theboundary was calculated in the previous subsection In the(formal) path integral (2) we will ignore all problems (ill-definiteness singularities etc) of the path integral approachNext we have to discuss the measure 119863119890 of the path integralCurrently there is no rigorous definition of this measure andas usual we assume a product measure

Then we have two possible parts which are more or lessindependent from each other

(i) integration119863119890119866over geometries

(ii) integration119863119890119863119878

over different differential structuresparametrized by some structure (see below)

Now we have to consider the following path integral

119885 (119872)

= intDiff structures

119863119890119863119878(int

Geometries119863119890

119866exp ( 119894

ℎ119878119864119867 [119890119872]))

(16)

and we have to calculate the influence of the differentialstructures first At this level we need an example an exotic1198783timesR

33 Constructing Exotic 1198783 times R In [28] Freedman con-structed the first example of an exotic 1198783 times R of special typeThere are also uncountable many different exotic R4 havingan end homeomorphic to 1198783 times R but not diffeomorphic toit But Freedmanrsquos first example is not of this type (as anend of an exotic R4) Therefore to get an infinite number ofdifferent exotic 1198783 times R one has to see 1198783 times R as an end ofR4 also expressible as complement R4

1198634 of the 4-disk A

second possibility is the usage of the end-sum technique ofGompf so that the standard 1198783 times R can be transformed intoan exotic 1198783 times R by end-sum with an exoticR4 Here we willconcentrate on the first construction that is the exotic 1198783timesRis an end of an exotic R4

Furthermore we will restrict on a subclass of exotic R4

called small exotic R4 (exotic R4 which can be embed-ded in a 4-sphere 1198784) For this class there is an explicithandle decomposition Small exotic R4rsquos are the result ofan anomalous behavior in 4-dimensional topology In 4-manifold topology [2] a homotopy equivalence between twocompact closed simply connected 4-manifolds implies ahomeomorphism between them (the so-called h cobordism)But Donaldson [29] provided the first smooth counterex-ample that this homeomorphism is not a diffeomorphism


that is both manifolds are generally not diffeomorphic toeach other The failure can be localized at some contractiblesubmanifold (Akbulut cork) so that an open neighborhood ofthis submanifold is a small exotic R4 The whole procedureimplies that this exotic R4 can be embedded in the 4-sphere1198784 The idea of the construction is simply given by the factthat every smooth h-cobordism between nondiffeomorphic4-manifolds can be written as a product cobordism exceptfor a compact contractible sub-h-cobordism 119881 the Akbulutcork An open subset 119880 sub 119881 homeomorphic to [0 1] times R4

is the corresponding sub-h-cobordism between two exoticR4rsquos These exotic R4rsquos are called ribbon R4rsquos They havethe important property of being diffeomorphic to opensubsets of the standardR4 In [30] Freedman andDeMichelisconstructed also a continuous family of small exoticR4 Nowwe are ready to discuss the decomposition of a small exoticR4 by Bi zaca andGompf [31] using special pieces the handlesforming a handle body Every 4-manifold can be decomposed(seen as handle body) using standard pieces such as119863119896

times1198634minus119896

the so-called 119896-handle attached along 120597119863119896times 119863

4minus119896 to theboundary 1198783 = 120597119863

4 of a 0-handle 1198630times 119863

4= 119863

4 Theconstruction of the handle body for the small exoticR4 called1198774 in the following can be divided into two parts

1198774= 119860cork⋃

1198632

CH decomposition of small exotic R4

(17)

The first part is known as the Akbulut cork a contractible 4-manifold with boundary a homology 3-sphere (a 3-manifoldwith the same homology as the 3-sphere) The Akbulut cork119860cork is given by a linking between a 1-handle and a 2-handleof framing 0The second part is the Casson handle CHwhichwill be considered now

Let us start with the basic construction of the Cassonhandle CH Let119872 be a smooth compact simple-connected4-manifold and 119891 1198632

rarr 119872 a (codimension-2) mappingBy using diffeomorphisms of 1198632 and119872 one can deform themapping 119891 to get an immersion (ie injective differential)generically with only double points (ie |119891minus1

(119891(119909))| = 2)as singularities [32] But to incorporate the generic locationof the disk one is rather interesting in the mapping of a 2-handle 1198632

times 1198632 induced by 119891 times 119894119889 1198632

times 1198632rarr 119872 from 119891

Then every double point (or self-intersection) of 119891(1198632) leads

to self-plumbings of the 2-handle 1198632times 119863

2 A self-plumbingis an identification of1198632

0times119863

2 with1198632

1times119863

2 where1198632

0 119863

2

1sub

1198632 are disjoint subdisks of the first factor disk In complex

coordinates the plumbing may be written as (119911 119908) 997891rarr (119908 119911)

or (119911 119908) 997891rarr (119908 119911) creating either a positive or negative(resp) double point on the disk 1198632

times 0 (the core) Considerthe pair (1198632

times 1198632 120597119863

2times 119863

2) and produce finitely many self-

plumbings away from the attaching region 1205971198632times 119863

2 to get akinky handle (119896 120597minus119896) where 120597minus119896 denotes the attaching regionof the kinky handle A kinky handle (119896 120597minus119896) is a one-stagetower (119879

1 120597

minus1198791) and an (119899+1)-stage tower (119879

119899+1 120597

minus119879119899+1) is an

119899-stage tower union kinky handles cup119899ℓ=1(119879

ℓ 120597

minus119879ℓ) where two

towers are attached along 120597minus119879ℓ Let 119879minus

119899be (interior119879

119899)cup120597

minus119879119899

and the Casson handle

CH = ⋃

ℓ=0

119879minus

ℓ (18)

is the union of towers (with direct limit topology inducedfrom the inclusions 119879

119899997893rarr 119879

119899+1)

Themain idea of the construction above is very simple animmersed disk (disk with self-intersections) can be deformedinto an embedded disk (disk without self-intersections) bysliding one part of the disk along another (embedded) diskto kill the self-intersections Unfortunately the other disk canbe immersed only But the immersion can be deformed toan embedding by a disk again and so forth In the limit ofthis process one ldquoshifts the self-intersections into infinityrdquo andobtains the standard open 2-handle (1198632

times R2 120597119863

2times R2

) Inthe proof of Freedman [2] themain complications come fromthe lack of control about this process

A Casson handle is specified up to (orientation preserv-ing) diffeomorphism (of pairs) by a labeled finitely branchingtree with base-point lowast having all edge paths infinitelyextendable away from lowast Each edge should be given a label+ or minus Here is the construction tree rarr CH Each vertexcorresponds to a kinky handle the self-plumbing number ofthat kinky handle equals the number of branches leaving thevertex The sign on each branch corresponds to the sign ofthe associated self-plumbing The whole process generates atree with infinitely many levels In principle every tree with afinite number of branches per level realizes a correspondingCasson handle Each building block of a Casson handle theldquokinkyrdquo handle with 119899 kinks is diffeomorphic to the 119899-timesboundary connected sum

119899(119878

1times119863

3) (see Appendix A)with

two attaching regions The number of end connected sums isexactly the number of self-intersections of the immersed twohandle One region is a tubular neighborhood of band sumsof Whitehead links connected with the previous block Theother region is a disjoint union of the standard open subsets1198781times 119863

2 in 1198991198781times 119878

2= 120597(

1198991198781times 119863

3) (this is connected with

the next block)For the construction of an exotic 1198783 times R denoted

by 1198783times120579R we consider the complement 1198774

1198634 or the

decomposition

1198783times120579R = 119877

4 119863

4= (119860cork 119863

4)⋃

1198632

CH (19)

The first part119860cork 1198634 contains a cobordism between the

3-sphere 1198783 and the boundary of the Akbulut cork 120597119860cork (ahomology 3-sphere)The complementR4

1198634 is conformally

equivalent to 1198783 times R Equivalently the complement 1198774 119863

4

is diffeomorphic to an exotic 1198783 timesR But the exoticness is notconfined to a compact subset but concentrated at infinity (forinstance at +infin) In our case we choose a decomposition like

1198783times120579R = 119872(119878

3 120597119860cork)⋃

1198632

CH (20)

where 119872(1198783 120597119860cork) is a cobordism between 1198783 and 120597119860corkFor the Casson handle we need another representation


obtained by using Morse theory (see [33]) Every kinkyhandle (119896 120597minus119896) is given by 119899 pairs of 1-2-handle pairs where119899 is the number of kinks (or self-intersections)These handlesare given by the level sets of the Morse functions

1198911= 119909

2+ 119910

2+ 119911

2minus 119905

2 for the 1-handle

1198912= 119909

2+ 119910

2minus 119911

2minus 119905

2 for the 2-handle(21)

that is by the sets 119871(119891119894 119862) = (119909 119910 119911 119905) | 119891

119894(119909 119910 119911 119905) = 119862 =

const for 119894 = 1 2 Now we represent the Casson handle bythe union of all 119899-stage towers

CH = ⋃

level ℓ of treeT(int (119879

ℓ) cup 120597

minus119879ℓ) (22)

arranged along the treeT But every tower 119879ℓis given by the

union of pairs (1198911 119891

2) But what is the geometry of 119879

ℓ(and

better of int(119879ℓ)) Every level set 119871(119891

1 119862) and 119871(119891

2 119862) is a

hyperbolic 3-manifold (ie with negative curvature) and theunion of all level sets is a hyperbolic 4-manifold A centralpoint in our argumentation is Mostow rigidity a centralproperty of all hyperbolic 3-manifolds (or higher) with finitevolume explained in the next subsection

34 The Hyperbolic Geometry of 119862119867 The central element inthe Casson handle is a pair of 1- and 2-handles representinga kinky handle As we argued above this pair admits ahyperbolic geometry (or it is a hyperbolic 3-manifold) havingnegative scalar curvature A 3-manifold admits a hyperbolicstructure in the interior if there is a diffeomorphism toH3Γ where Γ is a discrete subgroup Γ sub SO(3 1) of

the Lorentz group and we have a representation of thefundamental group 120587

1(119872) into SO(3 1) (the isometry group

of the hyperbolic space H3) One property of hyperbolic 3-and 4-manifolds is central Mostow rigidity As shown byMostow [34] every hyperbolic 119899-manifold 119899 gt 2 with finitevolume has this property Every diffeomorphism (especiallyevery conformal transformation) of a hyperbolic 119899-manifoldwith finite volume is induced by an isometry Therefore onecannot scale a hyperbolic 3-manifold with finite volumeThen the volume vol() and the curvature are topologicalinvariants but for later usages we combine the curvature andthe volume into the Chern-Simons invariant CS() But moreis true in a hyperbolic 3-manifold there are special surfaceswhich cannot be contracted called incompressible surfaceA properly embedded connected surface 119878 sub 119873 in a 3-manifold119873 is called 2-sided if its normal bundle is trivial and1-sided if its normal bundle is nontrivialThe ldquosidesrdquo of 119878 thencorrespond to the components of the complement of 119878 in atubular neighborhood 119878 times [0 1] sub 119873 A 2-sided connectedsurface 119878 other than 1198782 or 1198632 is called incompressible iffor each disk 119863 sub 119873 with 119863 cap 119878 = 120597119863 there is a disk119863

1015840sub 119878 with 1205971198631015840

= 120597119863 that is the boundary of the disk119863 can be contracted in the surface 119878 The boundary of a 3-manifold is an incompressible surfaceMore importantly thissurface can be detected in the fundamental group 120587

1(119873) of

the 3-manifold that is there is an injective homomorphism1205871(119878) rarr 120587

1(119873) The consequence of all properties is the

following conclusion

The tower 119879ℓhas a hyperbolic geometry (with finite

volume) and therefore fixed size that is it cannot be scaledby any diffeomorphism or conformal transformation Thenwe obtain an invariant decomposition of the Casson handleinto towers arrangedwith respect to a tree Secondly inside ofevery tower119879

ℓthere is (at least one) an incompressible surface

also of fixed sizeIn case of the tower 119879

ℓ one knows two incompressible

surfaces the two tori coming from the complement ofthe Whitehead link (with two components) used in theconstruction

35The Path Integral of the Exotic 1198783timesR Nowwewill discussthe path integral using the decomposition

1198783times120579R

= 119872(1198783 120597119860cork)⋃

1198632

( ⋃


ℓ) cup 120597

minus119879ℓ))

(23)

and we remark that the construction of the cobordism119872(119878

3 120597119860cork) requires the usage of a Casson handle again

denoted by119872(1198783 120597119860cork)cupCHcorkTherefore we have to clar-ify the role of the Casson handle In the previous Section 34we discussed the strong connection between geometry andtopology for hyperbolic manifoldsThe topology of 1198783times

120579R is

rather trivial but the smoothness structure (and therefore thedifferential topology) can be very complicate

As stated above the boundary terms can be factorizedfrom the terms in the interior Formally we obtain

119885(1198783times120579R) = prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3)

times (prod

ℓ

119885 (int (119879ℓ))119885 (CHcork)

(24)

and for an expectation value of the observable O

⟨1198783times120579R |O| 119878

3times120579R⟩ (25)

but for the following we have to discuss it more fully Tounderstand the time-like evolution of a disk (or a surface) wehave to describe a disk inside of a Casson handle as pioneeredby Bi zaca [35] With the same arguments one can alsodescribe the modification of the 3-sphere into homology 3-spheres Σ But then we obtain (formally) an infinite sequenceof homology 3-spheres Σ

1rarr Σ

2rarr sdot sdot sdot with amplitudes

119885(1198783times120579R) = ⟨Σ

1

10038161003816100381610038161198791198721003816100381610038161003816 Σ2⟩ ⟨Σ2

10038161003816100381610038161198791198721003816100381610038161003816 Σ3⟩ sdot sdot sdot (26)

including the boundary terms Every spatial section Σ119899can

be seen as an element of the phase space in quantum gravityTherefore this change of transitions is a topological phasetransition which will be further investigated in our work

The choice of the boundary term has a kind of arbitrari-ness We can choose the decomposition much finer to get


more boundary terms Therefore the path integral (14) mustbe extended away from the boundary We will discuss thisextension also in our forthcoming work

Before we go ahead we have to discuss the foliationstructure of 1198783times

120579R or the appearance of different time

variables As stated above our space-time has the topologyof 1198783 timesR with equal slices parametrized by a topological time119905TOP that is

(1198783times120579R)

TOP= (119901 119905TOP) | 119901 isin 119878

3 119905TOP isin R

= 1198783times 119905TOP | 119905TOP isin minusinfin +infin

(27)

defined by the topological embedding 1198783 997893rarr 1198783times120579R It is

the defining property of exotic smoothness that 1198783 insideof 1198783times

120579R is only a topological 3-sphere that is it is wildly

embedded and so only represented by an infinite polyhedronThere is another possibility to introduce 119905TOP whichwill pointus to the smooth case For that purpose we define a map 119865 1198783timesR rarr R by (119909 119905) 997891rarr 119905 so that 119905TOP = 119865(119901) for 119901 isin 119878

3timesR

In contrast one also has the smooth time 119905Diff which we haveto define now Locally it is the smooth (physical coordinate)time We know also that the exotic 1198783times

120579R is composed by a

sequence Σ1rarr Σ

2rarr Σ

3rarr sdot sdot sdot of homology 3-spheres

or better by a sequence 119872(Σ1 Σ

2)cup

Σ2119872(Σ

2 Σ

3)cup

Σ3 of

(homology) cobordism between the homology 3-spheresAll sequences are ordered and so it is enough to analyzeone cobordism 119872(Σ

1 Σ

2) Every cobordism between two

homology 3-spheres Σ1and Σ

2is characterized by the

existence of a finite number of 1-2-handle pairs (or dually2-3-handle pairs) Now we define a smooth map 119865cob

119872(Σ1 Σ

2) rarr [0 1] which must be a Morse function (ie

it has isolated critical points) [33] The number of criticalpoints 119873 of 119865 is even say 119873 = 2119896 where 119896 is the numberof 1-2-handle pairs These critical points are also denoted asnaked singularities in GR (but of bounded curvature) Likein the case of topological time 119905TOP we introduce the smoothtime by 119905Diff = 119865cob(119901) for all 119901 isin 119872(Σ1 Σ2) sub 119878

3times120579R The

extension of 119905Diff to the whole 1198783times120579R by the Morse function

119865 1198783times120579R rarr R is straightforward 119905Diff = 119865(119901) for all 119901 isin

1198783times120579R The cobordism119872(Σ

1 Σ

2) is part of the exotic 1198783times

120579R

and can be embedded to make it 1198783 times [0 1] topologicallyTherefore function 119865cob is a continuous function which isstrictly increasing on future directed causal curves so it is atime function (see [36 37]) But there is also another methodto construct 119905Diff by using codimension-1 foliations In [7] weuncovered a strong relation between codimension-1 foliations(also used to construct a Lorentz structure on a manifold)and exotic smoothness structures for a small exotic R4 Thecoordinate of this codimension-1 submanifold is also thesmooth time 119905Diff This approach will be more fully discussedin our forthcoming work

4 The Expectation Value ofthe Area and Inflation

In Section 34 we described the hyperbolic geometry origi-nated in the exotic smoothness structure of 1198783times

120579R Because

of this hyperbolic geometry there are incompressible surfacesinside of the hyperbolic manifold as the smallest possibleunits of geometry Then Mostow rigidity determines thebehavior of this incompressible surface At first we willconcentrate on the first cobordism119872(119878

3 120597119860cork) between 119878

3

and the boundary 120597119860cork of the Akbulut cork The area of asurface is given by

119860 (119890 119878) = int119878

1198892120590radic119864119886119864119887119899

119886119899119887 (28)

with the normal vector 119899119886and the densitized frame 119864119886 =

det(119890)119890119886 The expectation value of the area 119860

⟨1198783|119860 (119890 119878)| 120597119860cork⟩

=1

119885 (119872(1198783 120597119860cork))

times int119863119890119860 (119890 119878) exp ( 119894ℎ119878119864119867[119890119872 (119878

3 120597119860cork])

(29)

depends essentially on the hyperbolic geometry As arguedabove this cobordism has a hyperbolic geometry but inthe simplest case the boundary of the Akbulut cork isthe homology 3-sphere 120597119860cork = Σ(2 5 7) a Brieskornhomology 3-sphere Now we study the area of a surfacewhere one direction is along the time axis Then we obtaina decomposition of the surface into a sum of small surfacesso that every small surface lies in one component of thecobordism Remember that the cobordism 119872(119878

3 120597119860cork) is

decomposed into the trivial cobordism 1198783times [0 1] and a

Casson handle CH = cupℓ119879ℓ Then the decomposition of the

surface

119878 = cupℓ119878ℓ (30)

corresponds to the decomposition of the expectation value ofthe area

119860ℓ(119890 119878

ℓ) = int

119878ℓ

1198892120590radic119864119886119864119887119899

119886119899119887 (31)

so that

⟨120597minus119879ℓ

1003816100381610038161003816119860ℓ1015840(119890 119878

ℓ1015840)1003816100381610038161003816 120597

minus119879ℓ+1⟩ = ⟨120597

minus119879ℓ

1003816100381610038161003816119860ℓ(119890 119878

ℓ)1003816100381610038161003816 120597

minus119879ℓ+1⟩ 120575

ℓℓ1015840

⟨1198783|119860 (119890 119878)| 120597119860cork⟩ = sum

ℓ

⟨120597minus119879ℓ

1003816100381610038161003816119860ℓ(119890 119878

ℓ)1003816100381610038161003816 120597

minus119879ℓ+1⟩

(32)

The initial value for ℓ = 0 is the expectation value

⟨120597minus1198790

10038161003816100381610038161198600(119890 119878

0)1003816100381610038161003816 120597

minus1198791⟩ = 119886

2

0 (33)

where 1198860is the radius of the 3-sphere 1198783 But because of

the hyperbolic geometry (with constant curvature because ofMostow rigidity) every further level scales this expectationvalue by a constant factor Therefore to calculate the expec-tation value we have to study the scaling behavior


Consider a cobordism119872(Σ0 Σ

1) between the homology

3-spheres Σ0 Σ

1 As shown by Witten [38ndash40] the action

intΣ01

3

119877radicℎ1198893119909 = 119871 sdot CS (Σ

01) (34)

for every 3-manifold (in particular for Σ0and Σ

1denoted by

Σ01) is related to the Chern-Simons action CS(Σ

01) (defined

in Appendix B) The scaling factor 119871 is related to the volumeby 119871 = 3radicvol(Σ

01) and we obtain formally

119871 sdot CS (Σ01 119860) = 119871

3sdotCS (Σ

01)

1198712= int

Σ01

CS (Σ01)

1198712radicℎ119889

3119909

(35)

by using

1198713= vol (Σ

01) = int

Σ01

radicℎ1198893119909 (36)

Together with

3119877 =

3119896

1198862 (37)

one can compare the kernels of the integrals of (34) and (35)to get for a fixed time

3119896

1198862=CS (Σ

01)

1198712 (38)

This gives the scaling factor

120599 =1198862

1198712=

3

CS (Σ01) (39)

where we set 119896 = 1 in the followingThe hyperbolic geometryof the cobordism is best expressed by the metric

1198891199042= 119889119905

2minus 119886(119905)

2ℎ119894119896119889119909

119894119889119909

119896 (40)

also called the Friedmann-Robertson-Walker metric (FRWmetric) with the scaling function 119886(119905) for the (spatial) 3-manifold But Mostow rigidity enforces us to choose

(119886

119886)

2

=1

1198712 (41)

in the length scale 119871 of the hyperbolic structure But whyis it possible to choose the FRW metric At first we statethat the FRW metric is not sensitive to the topology ofthe space-time One needs only a space-time which admitsa slicing with respect to a smooth time 119905Diff and a metricof constant curvature for every spatial slice Then for thecobordism119872(Σ

1 Σ

2) between Σ

1and Σ

2we have two cases

the curvature parameter 119896(Σ1) of Σ

1(say 119896(Σ

1) = +1)

jumps to the value 119896(Σ2) of Σ

2(say 119896(Σ

2) = minus1) or both

curvatures remain constantThe second case is the usual oneEach homology 3-sphere Σ

1 Σ

2has the same geometry (or

geometric structure in the sense of Thurston [41]) which is

hyperbolic in most case The first case is more complicatedHere we need the smooth function to represent the jumpin the curvature parameter 119896 Let us choose the function119896 R rarr R

119905 997891997888rarr +1 0 le 119905

1 minus 2 sdot exp (minus120582 sdot 119905minus2) 119905 gt 0(42)

which is smooth and the parameter 120582 determines the slope ofthis function Furthermore the metric (40) is also the metricof a hyperbolic space (which has to fulfill Mostow rigiditybecause the cobordism119872(Σ

1 Σ

2) is compact)

In the following we will switch to quadratic expressionsbecause we will determine the expectation value of the areaThen we obtain

1198891198862=1198862

1198712119889119905

2= 120599119889119905

2 (43)

with respect to the scale 120599 By using the tree of the Cassonhandle we obtain a countable infinite sum of contributionsfor (43) Before we start we will clarify the geometry of theCasson handle The discussion of the Morse functions aboveuncovers the hyperbolic geometry of the Casson handle (seealso Section 34) Therefore the tree corresponding to theCasson handle must be interpreted as a metric tree withhyperbolic structure in H2 and metric 1198891199042 = (1198891199092 + 1198891199102)1199102The embedding of the Casson handle in the cobordism isgiven by the following rules

(i) The direction of the increasing levels 119899 rarr 119899 + 1 isidentified with 1198891199102 and 1198891199092 is the number of edgesfor a fixed level with scaling parameter 120599

(ii) The contribution of every level in the tree is deter-mined by the previous level best expressed in thescaling parameter 120599

(iii) An immersed disk at level 119899 needs at least one disk toresolve the self-intersection pointThis disk forms thelevel 119899 + 1 but this disk is connected to the previousdisk So we obtain for 1198891198862|

119899+1at level 119899 + 1

1198891198862|119899+1

sim 120599 sdot 119889119886210038161003816100381610038161003816119899

(44)

up to a constantBy using the metric 1198891199042 = (119889119909

2+ 119889119910

2)119910

2 with theembedding (1199102 rarr 119899+1 1198891199092 rarr 120599) we obtain for the change119889119909

2119910

2 along the 119909-direction (ie for a fixed 119910) 120599(119899+1)Thischange determines the scaling from the level 119899 to 119899 + 1 thatis

119889119886210038161003816100381610038161003816119899+1

=120599

119899 + 1sdot 119889119886

210038161003816100381610038161003816119899=

120599119899+1

(119899 + 1)sdot 119889119886

2100381610038161003816100381610038160 (45)

and after the whole summation (as substitute for an integralfor the discrete values) we obtain for the relative scaling

1198862=

infin

sum

119899=0

(119889119886210038161003816100381610038161003816119899) = 119886

2

0sdot

infin

sum

119899=0

1

119899120599119899

= 1198862

0sdot exp (120599) = 1198862

0sdot 119897scale

(46)


with 1198891198862|0= 119886

2

0 With this result in mind we consider the

expectation value where we use the constant scalar curvature(Mostow rigidity) By using the normalization many termsare neglected (like the boundary terms)

⟨1198783|119860 (119890 119878)| 120597119860cork⟩

= ((prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3)

times

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩)

times(prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3))

minus1

)

=

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

(47)

Finally we obtain for the area 11988620for the first level ℓ = 0

⟨1198783|119860 (119890 119878)| 120597119860cork⟩ =

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

= 1198862

0sdot

infin

sum

119899=0

1

119899(

3

CS (120597119860cork))

119899

= 1198862

0sdot exp( 3

CS (120597119860cork))

(48)

with the radius 1198860of Σ

0and arrive at 119886 for Σ

1 From the

physical point of view we obtain an exponential increaseof the area that is we get an inflationary behavior Thisderivation can be also extended to the next Casson handlebut we have to determine the 3-manifold in which 120597119860cork canchange It will be done below

5 An Effective Theory

Now will ask for an effective theory where the influence ofthe exotic smoothness structure is contained in some moduli(or some field) As explained above the main characteris-tics is given by a change of the (spatial) 4-manifold (butwithout changing the homology) Therefore let us describethis change (the so-called homology cobordism) betweentwo homology 3-spheres Σ

0and Σ

1 The situation can be

described by a diagram

Σ1

Ψ

997888rarr R

120601 darr 119894119889

Σ0

120595

997888rarr R

(49)

which commutes The two functions 120595 and Ψ are the Morsefunction of Σ

0and Σ

1 respectively with Ψ = 120595 ∘ 120601 The

Morse function over Σ01

is a function Σ01rarr R having only

isolated nondegenerated critical points (ie with vanishing

first derivatives at these points) Ahomology 3-sphere has twocritical points (located at the two poles) The Morse functionlooks like plusmn1199092 at these critical points The transition 119910 =

120601(119909) represented by the (homology) cobordism 119872(Σ0 Σ

1)

maps the Morse function 120595(119910) = 1199102 on Σ

0to the

Morse function Ψ(119909) = 120601(119909)2 on Σ1 The function minus1206012

represents also the critical point of the cobordism119872(Σ0 Σ

1)

But as we learned above this cobordism has a hyperbolicgeometry and we have to interpret the function 120601(119909)2 notas Euclidean form but change it to the hyperbolic geometryso that

minus10038171003817100381710038171206011003817100381710038171003817

2= minus (120601

2

1+ 120601

2

2+ 120601

2

3) 997888rarr minus119890

minus21206011 (1 + 1206012

2+ 120601

2

3) (50)

that is we have a preferred direction represented by a singlescalar field 120601

1 Σ

1rarr R Therefore the transition Σ

0rarr

Σ1is represented by a single scalar field 120601

1 Σ

1rarr R and

we identify this field as the moduli Finally we interpret thisMorse function in the interior of the cobordism119872(Σ

0 Σ

1) as

the potential (shifted away from the point 0 ) of the scalar field120601 with Lagrangian

119871 = 119877 + (120597120583120601)

2

minus120588

2(1 minus exp (minus120582120601))2 (51)

with two free constants 120588 and 120582 For the value 120582 = radic23

and 120588 = 31198722 we obtain the Starobinski model [42] (by a

conformal transformation using 120601 and a redefinition of thescalar field [43])

119871 = 119877 +1

611987221198772 (52)

with the mass scale 119872 ≪ 119872119875much smaller than the

Planck mass From our discussion above the appearance ofthis model is not totally surprising It favors a surface to beincompressible (which is compatible with the properties ofhyperbolic manifolds) In the next section we will determinethis mass scale

6 A Cosmological Model Compared tothe Planck Satellite Results

In this section we will go a step further and discuss thepath integral for 1198783 times R where we sum over all smoothnessstructures Furthermore we will assume that 1198783 times R is theend of a small exotic R4 But then we have to discussthe parametrization of all Casson handles As discussedby Freedman [2] all Casson handles can be parametrizedby a dual tree where the vertices are 5-stage towers (withthree extra conditions) We refer to [2] or to [44] for thedetails of the well-known constructionThis tree has one rootfrom which two 5-stage towers branch Every tower has anattaching circle of any framing Using Bizacas technique [35]we obtain an attaching of a 5-tower along the sum 119875119875 oftwo Poincare spheres119875 (for the two towers)Therefore for theuniversal case we obtain two transitions

1198783 cork997888rarr 120597119860cork

tower997888rarr 119875119875 (53)


with the scaling behavior

119886 = 1198860sdot exp( 3

2 sdot CS (120597119860cork)+

3

2 sdot CS (119875119875)) (54)

It can be expressed by the expectation value

⟨1198783|119860 (119890 119878)| 119875119875⟩ = 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(55)

for the transition 1198783 rarr 119875119875 It is important to note that thisexpectation value is the sum over all smoothness structuresof 1198783 timesR and we obtain also

⟨1198783timesR |119860 (119890 119878)| 119878

3timesR⟩

= sum

diff structures⟨119878

3times120579R |119860 (119890 119878)| 119878

3times120579R⟩

= 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(56)

with 11988620as the size of the 3-sphere 1198783 at minusinfin With the

argumentation above the smoothness structure has a kind ofuniversality so that the two transitions above are generic

In our model (using the exotic smoothness structure) weobtain two inflationary phases In the first phase we have atransition

1198783997888rarr 120597119860cork (57)

and for the simplest case 120597119860cork = Σ(2 5 7) a Brieskornhomology 3-sphereNowwewill assume that the 3-sphere hasPlanck-size

1198860= 119871

119875= radic

ℎ119866

1198883 (58)

then we obtain for the size

119886 = 119871119875sdot exp( 3

2 sdot CS(Σ (2 5 7)) (59)

We can use the method of Fintushel and Stern [45ndash47] to cal-culate theChern-Simons invariants for the Brieskorn spheresThe calculation can be found in Appendix C Note that therelation (34) is only true for the Levi-Civita connectionThenthe Chern-Simons invariant is uniquely defined to be theminimum denoted by 120591() (see (B4)) Then we obtain for theinvariant (C7) so that

119871119875sdot exp(140

3) asymp 75 sdot 10

minus15119898 (60)

is the size of the cosmos at the end of the first inflationaryphaseThis size can be related to an energy scale by using it asCompton length and one obtains 165MeV comparable to theenergy scale of the QCD For the two inflationary transitions

1198783997888rarr Σ (2 5 7) 997888rarr 119875119875 (61)

one obtains the size

119886 = 119871119875sdot exp(140

3+ 90) asymp 914 sdot 10

24119898 asymp 10

9119871119895 (62)

As explained above the effective theory is the Starobinskymodel This model is in very good agreement with results ofthe Planck satellite [48] with the two main observables

119899119904sim 096 spectral index for scalar perturbations

119903 sim 0004 tensor-to-scalar ratio

but one parameter of the model is open the energy scale119872 in Planck units In our model it is related to the secondderivative of the Morse function which is the curvature ofthe critical point In our paper [49] we determined also theenergy scale of the inflation by using a simple argument toincorporate only the first 3 levels of the Casson handle Forthe scale

120599 =3

2 sdot CS (Σ (2 5 7)) (63)

of the first transition we obtain the scaling of the Planckenergy (associated with the Planck-sized 3-sphere at thebeginning)

119864Inflation =119864Planck

(1 + 120599 + (12059922) + (12059936)) (64)

with the relative scaling

120572 =119864Inflation119864Planck

=1

(1 + 120599 + (12059922) + (12059936))asymp 55325 sdot 10

minus5

(65)

leading to the energy scale of the inflation

119864Inflation asymp 67547 sdot 1014119866119890119881 (66)

by using 119864Planck asymp 12209 sdot 1019119866119890119881 We remark that the

relative scaling 120572 asymp 55325 sdot 10minus5 (in Planck units) above isan energy scale for the potential in the effective theory (52)Therefore we have to identify119872 = 120572 in the parameter 161198722

of the Starobinsky model (in very good agreement with themeasurements) Nowwe can go a step further and discuss theappearance of the cosmological constant

Again we can use the hyperbolic geometry to state thatthe curvature is negative and we have the Mostow rigiditythat is the scalar curvature of the 4-manifold has a constantvalue the cosmological constant Λ If we assume that the 3-sphere has the size of the Planck length (as above) then weobtain

Λ =1

1198712

119875

sdot exp(minus 3

CS (Σ (2 5 7))minus

3

CS (119875119875)) (67)

With the values of the Chern-Simons invariants (C7) weobtain the value

Λ sdot 1198712

119875= exp(minus280

3minus 180) asymp 5 sdot 10

minus118 (68)


in Planck units In cosmology one usually relate the cosmo-logical constant to the Hubble constant 119867

0(expressing the

critical density) leading to the length scale

1198712

119888=

1198882

31198672

0

(69)

The corresponding variable is denoted byΩΛand we obtain

ΩΛ=

1198885

3ℎ1198661198672

0

sdot exp(minus 3


3

CS (119875119875)) (70)

in units of the critical density This formula is in very goodagreement with theWMAP results that is by using the valuefor the Hubble constant

1198670= 74

km119904 sdotMpc

(71)

we are able to calculate the dark energy density to be

ΩΛ= 0729 (72)

agreeing with the WMAP results But it differs from thePlanck results [50] of the Hubble constant

(1198670)Planck = 68

km119904 sdotMpc

(73)

for which we obtain

ΩΛasymp 088 (74)

in contrast with the measured value of the dark energy

(ΩΛ)Planck = 0683 (75)

But there is another possibility for the size of the 3-sphere atthe beginning and everything depends on this choice But wecan use the entropy formula of a Black hole in Loop quantumgravity

119878 =119860 sdot 120574

0

4 sdot 120574 sdot 1198712

119875

sdot 2120587 (76)

with

1205740=

ln (2)120587 sdot radic3

(77)

according to [51] with the Immirzi parameter 120574 where theextra factor 2120587 is given by a different definition of 119871

119875

replacing ℎ by ℎ In the original approach of Ashtekar in Loopquantum gravity one usually set 120574 = 1 If we take it seriouslythen we obtain a reduction of the length in (70)

1

1198712

119875

997888rarr1

1198712

119875

sdot2 sdot ln (2)radic3

=2120587120574

0

1198712

119875

asymp 080037 sdot1

1198712

119875

(78)

or the new closed formula

ΩΛ=

1198885

3ℎ1198661198672

0

sdot 1205740sdot exp(minus 3


3

CS (119875119875))

(79)

correcting the valueΩΛasymp 088 to

ΩΛasymp 0704 (80)

But 1205740depends on the gauge group and if one uses the value

[52]

1205740=


(81)

agreeing also with calculations in the spin foam models [53]then one gets a better fit

ΩΛasymp 06836 (82)

which is in good agreement with the measurements

7 Conclusion

The strong relation between hyperbolic geometry (of thespace-time) and exotic smoothness is one of the mainresults in this paper Then using Mostow rigidity geometricobservables like area and volume or curvature are topologicalinvariants which agree with the expectation values of theseobservables (calculated via the path integral) We comparedthe results with the recent results of the Planck satellite andfound a good agreement In particular as a direct resultof the hyperbolic geometry the inflation can be effectivelydescribed by the Starobinski model Furthermore we alsoobtained a cosmological model which produces a realisticcosmological constant

Appendices

A Connected and Boundary ConnectedSum of Manifolds

Now we will define the connected sum and the boundaryconnected sum of manifolds Let119872119873 be two 119899-manifoldswith boundaries 120597119872 120597119873 The connected sum 119872119873 is theprocedure of cutting out a disk119863119899 from the interior int(119872) 119863

119899 and int(119873)119863119899 with the boundaries 119878119899minus1 ⊔120597119872 and 119878119899minus1 ⊔120597119873 respectively and gluing them together along the com-mon boundary component 119878119899minus1 The boundary 120597(119872119873) =120597119872 ⊔ 120597119873 is the disjoint sum of the boundaries 120597119872 120597119873 Theboundary connected sum119872119873 is the procedure of cutting outa disk119863119899minus1 from the boundary 120597119872119863

119899minus1 and 120597119873119863119899minus1 andgluing them together along 119878119899minus2 of the boundary Then theboundary of this sum119872119873 is the connected sum 120597(119872119873) =

120597119872120597119873 of the boundaries 120597119872 120597119873

B Chern-Simons Invariant

Let 119875 be a principal 119866 bundle over the 4-manifold 119872 with120597119872 = 0 Furthermore let 119860 be a connection in 119875 with thecurvature

119865119860= 119889119860 + 119860 and 119860 (B1)


and Chern class

1198622=

1

81205872int119872

tr (119865119860and 119865

119860) (B2)

for the classification of the bundle 119875 By using the Stokestheorem we obtain

int119872

tr (119865119860and 119865

119860) = int

120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B3)

with the Chern-Simons invariant

CS (120597119872119860) = 1

81205872int120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B4)

Now we consider the gauge transformation 119860 rarr 119892minus1119860119892 +

119892minus1119889119892 and obtain

CS (120597119872 119892minus1119860119892 + 119892minus1119889119892) = CS (120597119872119860) + 119896 (B5)

with the winding number

119896 =1

241205872int120597119872

(119892minus1119889119892)

3

isin Z (B6)

of the map 119892 119872 rarr 119866 Thus the expression

CS (120597119872119860) mod 1 (B7)

is an invariant the Chern-Simons invariant Now we willcalculate this invariant For that purpose we consider thefunctional (B4) and its first variation vanishes

120575CS (120597119872119860) = 0 (B8)

because of the topological invariance Then one obtains theequation

119889119860 + 119860 and 119860 = 0 (B9)

that is the extrema of the functional are the connectionsof vanishing curvature The set of these connections up togauge transformations is equal to the set of homomorphisms1205871(120597119872) rarr SU(2) up to conjugation Thus the calculation

of the Chern-Simons invariant reduces to the representationtheory of the fundamental group into SU(2) In [45] theauthors define a further invariant

120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (B10)

for the 3-manifold Σ This invariants fulfills the relation

120591 (Σ) =1

81205872intΣtimesR

tr (119865119860and 119865

119860) (B11)

which is the minimum of the Yang-Mills action10038161003816100381610038161003816100381610038161003816

1

81205872intΣtimesR

tr (119865119860and 119865

119860)

10038161003816100381610038161003816100381610038161003816

le1

81205872intΣtimesR

tr (119865119860and lowast119865

119860) (B12)

that is the solutions of the equation 119865119860= plusmn lowast 119865

119860 Thus the

invariant 120591(Σ) of Σ corresponds to the self-dual and anti-self-dual solutions on Σ timesR respectively Or the invariant 120591(Σ) isthe Chern-Simons invariant for the Levi-Civita connection

C Chern-Simons Invariant ofBrieskorn Spheres

In [45ndash47] an algorithm for the calculation of the Chern-Simons invariant for the Brieskorn sphere Σ(119901 119902 119903) ispresented According to that result a representation 120572

1205871(Σ(119901 119902 119903) rarr SU(2) is determined by a triple of 3 numbers

⟨119896 119897 119898⟩with 0 lt 119896 lt 119901 0 lt 119897 lt 119902 0 lt 119898 lt 119903 and the furtherrelations

119897

119902+119898

119903lt 1 119897 mod 2 = 119898 mod 2

119896

119901+119897

119902+119898

119903gt 1

119896

119901minus119897

119902+119898

119903lt 1

119896

119901+119897

119902minus119898

119903lt 1

(C1)

Then the Chern-Simons invariant is given by

CS (120572) = 1198902

4 sdot 119901 sdot 119902 sdot 119903mod 1 (C2)

with

119890 = 119896 sdot 119902 sdot 119903 + 119897 sdot 119901 sdot 119903 + 119898 sdot 119901 sdot 119902 (C3)

Now we consider the Poincare sphere 119875 with 119901 = 2 119902 =

3 119903 = 5 Then we obtain

⟨1 1 1⟩CS = 1

120

⟨1 1 3⟩CS = 49

120

(C4)

and for the Brieskorn sphere Σ(2 5 7)

⟨1 1 3⟩CS = 81

280

⟨1 3 1⟩CS = 9

280

⟨1 2 2⟩CS = 169280

⟨1 2 4⟩CS = 249280

(C5)

In [45] the authors define a further invariant

120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (C6)

for a homology 3-sphere Σ For 119875 and Σ(2 5 7) one obtains

120591 (119875) =1

120 120591 (Σ (2 5 7)) =

9

280 (C7)

And we are done


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors acknowledged all critical remarks of the refereesincreasing the readability of the paper

References

[1] S W Hawking and G F R Ellis The Large Scale Structure ofSpace-Time CambridgeUniversity Press Cambridge UK 1994

[2] M H Freedman ldquoThe topology of four-dimensional mani-foldsrdquo Journal of Differential Geometry vol 17 pp 357ndash34541982

[3] SDonaldson ldquoAn application of gauge theory to the topology of4-manifoldsrdquo Journal of Differential Geometry vol 18 pp 279ndash315 1983

[4] T Asselmeyer-Maluga ldquoA chicken-and-egg problem whichcame first the quantum state or spacetimerdquo 2012 httpfqxiorgcommunityforumtopic1424

[5] Fourth Prize of the FQXi Essay contest ldquoQuestioning theFoundationsrdquo 2012 httpfqxiorgcommunityessaywinners20121

[6] H Pfeiffer ldquoQuantum general relativity and the classification ofsmooth manifoldsrdquo Tech Rep DAMTP 2004-32 2004

[7] T Asselmeyer-Maluga and J Krol ldquoAbelian gerbes generalizedgeometries and exotic 1198774rdquo httparxivorgabs09041276

[8] T Asselmeyer-Maluga and R Mader ldquoExotic 1198774 and quantumfield theoryrdquo Journal of Physics Conference Series vol 343Article ID 012011 2012

[9] T Asselmeyer-Maluga and J Krol ldquoExotic smooth 1198774 and

certain configurations of NS and D branes in string theoryrdquoInternational Journal of Modern Physics A vol 26 no 7-8 pp1375ndash1388 2011

[10] T Asselmeyer-Maluga and J Krol ldquoQuantum D-branes andexotic smooth R4rdquo International Journal of Geometric Methodsin Modern Physics vol 9 no 3 p 1250022 18 2012

[11] C H Brans ldquoLocalized exotic smoothnessrdquo Classical andQuantum Gravity vol 11 no 7 pp 1785ndash1792 1994

[12] C H Brans ldquoExotic smoothness and physicsrdquo Journal ofMathematical Physics vol 35 pp 5494ndash5506 1994

[13] J Sładkowski ldquoGravity on exotic R4 with few symmetriesrdquoInternational Journal of Modern Physics D vol 10 pp 311ndash3132001

[14] J Sładkowski ldquoExotic smoothness noncommutative geome-try and particle physicsrdquo International Journal of TheoreticalPhysics vol 35 no 10 pp 2075ndash2083 1996

[15] J Sładkowski ldquoExotic smoothness and particle physicsrdquo ActaPhysica Polonica B vol 27 pp 1649ndash1652 1996

[16] J Sładkowski ldquoExotic smoothness fundamental interactionsand noncommutative geometryrdquo httparxivabshep-th9610093

[17] T Asselmeyer-Maluga and J Krol ldquoTopological quantum D-branes and wild embeddings from exotic smooth 1198774rdquo Interna-tional Journal ofModern Physics A vol 26 no 20 pp 3421ndash34372011

[18] T Asselmeyer-Maluga and J Krol ldquoQuantum geometry andwild embeddings as quantum statesrdquo International Journalof Geometric Methods in Modern Physics vol 10 Article ID1350055 2013

[19] R Kirby and L C Siebenmann Foundational Essays on Topo-logical Manifolds Smoothings and Triangulations Annals ofMathematics Studies PrincetonUniversity Press PrincetonNJUSA 1977

[20] A Ashtekar J Engle and D Sloan ldquoAsymptotics and Hamil-tonians in a first-order formalismrdquo Classical and QuantumGravity vol 25 no 9 p 095020 2008

[21] A Ashtekar and D Sloan ldquoAction and Hamiltonians in higher-dimensional general relativity first-order frameworkrdquo Classicaland Quantum Gravity vol 25 no 22 p 225025 2008

[22] S W Hawking ldquoThe path-integral approach to quantumgravityrdquo in General Relativity An Einstein Centenary SurveyI Hawking Ed pp 746ndash789 Cambridge University PressCambridge UK 1979

[23] T Asselmeyer-Maluga and H Rose ldquoOn the geometrization ofmatter by exotic smoothnessrdquo General Relativity and Gravita-tion vol 44 no 11 pp 2825ndash2856 2012

[24] L Dabrowski and G Dossena ldquoDirac operator on spinors anddiffeomorphismsrdquo Classical and Quantum Gravity vol 30 no1 Article ID 015006 2013

[25] T Friedrich ldquoOn the spinor representation of surfaces ineuclidean 3-spacerdquo Journal of Geometry and Physics vol 28 no1-2 pp 143ndash157 1998

[26] EWitten ldquoGlobal gravitational anomaliesrdquo Communications inMathematical Physics vol 100 no 2 pp 197ndash229 1985

[27] T Yoshida ldquoThe 120578-invariant of hyperbolic 3-manifoldsrdquo Inven-tiones Mathematicae vol 81 pp 473ndash514 1985

[28] M H Freedman ldquoA fake 1198783 times 119877rdquo Annals of Mathematics vol110 pp 177ndash201 1979

[29] S Donaldson ldquoIrrationality and the h-cobordism conjecturerdquoJournal of Differential Geometry vol 26 pp 141ndash168 1987

[30] S DeMichelis and M H Freedman ldquoUncountable many exotic1198774rsquos in standard 4-spacerdquo Journal of Differential Geometry vol

35 pp 219ndash254 1992[31] Z Bizaca and R Gompf ldquoElliptic surfaces and some simple

exoticR4rsquos rdquo Journal of Differential Geometry vol 43 no 3 pp458ndash504 1996

[32] M Golubitsky and V Guillemin Stable Mappings and TheirSingularities vol 14 ofGraduate Texts inMathematics SpringerNew York NY USA 1973

[33] J MilnorMorseTheory vol 51 ofAnnals ofMathematical StudyPrinceton University Press Princeton NJ USA 1963

[34] G D Mostow ldquoQuasi-conformal mappings in n-space and therigidity of hyperbolic space formsrdquo Publications Mathematiquesde lrsquoInstitut des Hautes Etudes Scientifiques vol 34 pp 53ndash1041968

[35] Z Bizaca ldquoAn explicit family of exotic Casson handlesrdquoProceed-ings of the American Mathematical Society vol 123 no 4 pp1297ndash1302 1995

[36] A N Bernal and M Sanchez ldquoFurther results on the smootha-bility of Cauchy hypersurfaces and Cauchy time functionsrdquoLetters in Mathematical Physics vol 77 no 2 pp 183ndash197 2006

[37] A N Bernal and M Sanchez ldquoGlobally hyperbolic spacetimescan be defined as ldquocausalrdquo instead of lsquostrongly causalrsquordquo Classicaland Quantum Gravity vol 24 no 3 pp 745ndash750 2007

[38] E Witten ldquo2 + 1 dimensional gravity as an exactly solublesystemrdquo Nuclear Physics B vol 311 pp 46ndash78 1988


[39] EWitten ldquoTopology-changing amplitudes in 2+1 dimensionalgravityrdquo Nuclear Physics B vol 323 pp 113ndash140 1989

[40] E Witten ldquoQuantization of Chern-Simons gauge theorywith complex gauge grouprdquo Communications in MathematicalPhysics vol 137 pp 29ndash66 1991

[41] W Thurston Three-Dimensional Geometry and TopologyPrinceton University Press Princeton NJ USA 1st edition1997

[42] A A Starobinski ldquoA new type of isotropic cosmological modelswithout singularityrdquo Physical Letters B vol 91 pp 99ndash102 1980

[43] BWhitt ldquoFourth order gravity as general relativity plusmatterrdquoPhysical Letters B vol 145 pp 176ndash178 1984

[44] M Freedman and F Quinn Topology of 4-Manifolds PrincetonMathematical Series Princeton University Press Princeton NJUSA 1990

[45] R Fintushel and R J Stern ldquoInstanton homology of Seifertfibred homology three spheresrdquo Proceedings of the LondonMathematical Society vol 61 pp 109ndash137 1990

[46] P Kirk and E Klassen ldquoChern-Simons invariants of 3-manifolds and representation spaces of knot groupsrdquoMathema-tische Annalen vol 287 no 1 pp 343ndash367 1990

[47] D S Freed and R E Gompf ldquoComputer calculation of Wit-tenrsquos 3-manifold invariantrdquo Communications in MathematicalPhysics vol 141 pp 79ndash117 1991

[48] P Ade N Aghanim C Armitage-Caplan et al ldquoPlanck2013 results XXII Constraints on inflationrdquo httparxivorgabs13035082

[49] T Asselmeyer-Maluga and J Krol ldquoOn the origin of inflation byusing exotic smoothnessrdquo httparxivorgabs13013628

[50] P Ade N Aghanim C Armitage-Caplan et al ldquoPlanck 2013results XVI Cosmological parametersrdquo httparxivorgabs13035076

[51] A Ashtekar J Baez A Corichi and K Krasnov ldquoQuantumgeometry and black hole entropyrdquo Physical Review Letters vol80 no 5 pp 904ndash907 1998

[52] M Domagala and J Lewandowski ldquoBlack-hole entropy fromquantum geometryrdquoClassical and QuantumGravity vol 21 no22 pp 5233ndash5243 2004

[53] M H Ansari ldquoGeneric degeneracy and entropy in loop quan-tum gravityrdquo Nuclear Physics B vol 795 no 3 pp 635ndash6442008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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FluidsJournal of

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Advances in Condensed Matter Physics

OpticsInternational Journal of



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AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


is only one smooth atlas for 119872 all other possibilities canbe transformed into each other by a diffeomorphism But incontrast the deep results of Freedman [2] on the topologyof 4-manifolds combined with Donaldsonrsquos work [3] gave thefirst examples of nondiffeomorphic smoothness structureson 4-manifolds including the well-known R4 Much of themotivation can be found in the FQXI essay [4 5] Herewewilldiscuss another property of the exotic smoothness structureits quantum geometry in the path integral

Diffeomorphism invariance is the most important prop-erty of the Einstein-Hilbert action with far reaching conse-quences [6] One of our results is a close relation betweengeometry and foliation to exotic smoothness [7 8] In theparticular example of the exotic R4 we discussed the exoticsmoothness structures as a manifestation of quantum gravity(by using string theory [9 10]) This exotic R4 has someinteresting properties as first noted by Brans [11 12] Moreimportantly as shown by Sładkowski [13] the exotic R4 hasa nontrivial curvature in contrast to the flat standard R4It was the first result that an exotic R4 can be seen as asource of gravity (or it must contain sources of gravity)Sładkowski [14ndash16] went further and showed a relation toparticle physics also related to quantum gravity But whythere is a relation to quantum gravity In [17] we presentedthe first idea to understand this relation which was furtherextended in [18] An exotic 4-manifold like 1198783 times R is alsocharacterized by the property that there is no smoothlyembedded 3-sphere but a topological embedded one Thistopological 1198783 is wildly embedded that is the image of theembedding must be triangulated by an infinite polyhedronIn [18] we proved that the (deformation) quantization of ausual (or tame) embedding is a wild embedding which canbe seen as a quantum state But then any exotic 4-manifoldcan be interpreted as a quantum state of the 4-manifoldwith standard smoothness structure From this point ofview the calculation of the path integral in quantum gravityhas to include the exotic smoothness structures Usually itis hopeless to make these calculations But by using theclose relation of exotic smoothness to hyperbolic geometryone has a chance to calculate geometric expressions likethe expectation value of the surface area In this paper wewill show that this expectation value has an inflationarybehavior that is the area grows exponentially (along thetime axis) Therefore quantum gravity (in the sense of exoticsmoothness) can be the root of inflation

2 Space-Time and Smoothness

From the mathematical point of view the space-time isa smooth 4-manifold endowed with a (smooth) metric asbasic variable for general relativity The existence questionfor Lorentz structure and causality problems (see Hawkingand Ellis [1]) give further restrictions on the 4-manifoldcausality implies noncompactness Lorentz structure needsa nonvanishing normal vector field Both concepts can becombined in the concept of a global hyperbolic 4-manifold119872 having a Cauchy surface S so that119872 = S timesR

All these restrictions on the representation of space-timeby the manifold concept are clearly motivated by physicalquestions Among these properties there is one distinguishedelement the smoothness Usually one starts with a topolog-ical 4-manifold119872 and introduces structures on them Thenone has the following ladder of possible structures

Topology 997888rarr piecewise-linear (PL) 997888rarr Smoothness

997888rarr bundles Lorentz Spin etc

997888rarr metric geometry

(1)

We do not want to discuss the first transition that is theexistence of a triangulation on a topological manifold But weremark that the existence of a PL structure implies uniquelya smoothness structure in all dimensions smaller than 7 [19]Herewe have to consider the following steps to define a space-time

(i) Fix a topology for the space-time119872(ii) Fix a smoothness structure that is a maximal differ-

entiable atlasA(iii) Fix a smoothmetric or get one by solving the Einstein

equation

The choice of a topology never fixes the space-time uniquelythat is there are two space-times with the same topologywhich are not diffeomorphicThemain idea of the paper is theintroduction of exotic smoothness structures into space-timeIf two manifolds are homeomorphic but nondiffeomorphicthey are exotic to each other The smoothness structure iscalled an exotic smoothness structure

In dimension four there are many examples of com-pact 4-manifolds with countable infinite nondiffeomorphicsmoothness structures andmany examples of noncompact 4-manifolds with uncountable infinite many nondiffeomorphicsmoothness structures But in contrast the number of non-diffeomorphic smoothness structures is finite for any otherdimension [19] As an example we will consider the space-time 1198783 times R having uncountable many nondiffeomorphicsmoothness structures in the following

3 The Path Integral in Exotic 1198783 timesR

For simplicity we consider general relativity without matter(using the notation of topological QFT) Space-time is asmooth oriented 4-manifold 119872 which is noncompact andwithout boundary From the formal point of view (no diver-gences of the metric) one is able to define a boundary 120597119872 atinfinity The classical theory is the study of the existence anduniqueness of (smooth)metric tensors119892 on119872 that satisfy theEinstein equations subject to suitable boundary conditionsIn the first order Hilbert-Palatini formulation one specifiesan SO(1 3)-connection 119860 together with a cotetrad field 119890rather than a metric tensor Fixing 119860|

120597119872at the boundary

one can derive first-order field equations in the interior (nowcalled bulk) which are equivalent to the Einstein equationsprovided that the cotetrad is nondegenerate The theory isinvariant under space-time diffeomorphisms 119872 rarr 119872





119894119891= Σ

119894⊔ Σ

119891is the


119891to which




integral

⟨119860|Σ119891

10038161003816100381610038161198791198721003816100381610038161003816 119860|Σ119894

⟩

= int119860|120597119872119894119891

119863119860119863119890 exp( 119894ℎ119878119864119867[119890 119860119872

119894119891])

(2)



119894rarr H



119886





119892119886119887= 119890

119868

119886119890119869


119878119864119867[119890 119860119872

119894119891 120597119872

119894119891]

= int119872119894119891

120598119868119869KL (119890

119868and 119890

119869and (119889119860 + 119860 and 119860)

KL)

+ int120597119872119894119891

120598119868119869KL (119890

119868and 119890

119869and 119860

KL)

(3)


119868and

119890119869and119860



119894= Σ

119891are


119891and H = H

119894=




119885(1198721015840) = trH119879119872 = int119863119860119863119890 exp( 119894

ℎ119878119864119867 [119890 119860119872 120597119872])

(4)


119894119891 the


119872|119860|

Σ119894⟩ is the




⟨Σ119891

10038161003816100381610038161198791198721003816100381610038161003816 Σ119894⟩ = ⟨119860|Σ119891

10038161003816100381610038161198791198721003816100381610038161003816 119860|Σ119894

⟩ (5)


119894119891cupΣ119891119872

119891ℎ

with the amplitudes

⟨Σℎ

10038161003816100381610038161198791198721003816100381610038161003816 Σ119894⟩ = sum

119860|Σ119891

⟨Σℎ

10038161003816100381610038161198791198721003816100381610038161003816 Σ119891⟩ ⟨Σ119891

10038161003816100381610038161198791198721003816100381610038161003816 Σ119894⟩ (6)



119878120597[Σ

119891] = int

Σ119891

120598119868119869KL (119890

119868and 119890

119869and 119860

KL) = int

Σ119891

119867radicℎ1198893119909 (7)





rarr



119878plusmn




by 119878Σ= 120580

lowast119878+


lowastΦ for the


+

119872)) Let nabla119872

119883 nabla

Σ

119883be the


nabla119872

119883(Φ) = nabla

Σ

119883120601 minus

1

2(nabla

119883

rarr

119873) sdotrarr

119873 sdot120601 (8)




rarr


119883

rarr


119863119872Φ = 119863

3119863120601 minus 119867120601 (9)


119863119872Φ = 0 (10)


1198633119863120601 = 119867120601 (11)





1015840 Σ

1015840997893rarr 119872




119891 ∘ 119868 = 1198681015840∘ 119892 (12)


119892120590(Σ)


1(ΣZ




119892120590(Σ) is denoted by 119867

119892120590


119892120590to 119867

119891lowast119892119891lowast120590


119878120597[Σ

119891 ℎ] = int

Σ119891

119867radicℎ1198893119909 = int

Σ119891

120601 1198633119863120601119889

3119909 (13)




119885(Σ119891) = int119863ℎ exp( 119894

ℎ119878 [Σ

119891 ℎ])

= int119863120601119863120601 exp( 119894ℎintΣ119891

120601 1198633119863120601119889

3119909)


(14)






119878119864119867 [119890 119860119872] = int

119872

120598119868119869KL (119890

119868and 119890

119869and (119889119860 + 119860 and 119860)

KL) (15)




(ii) integration119863119890119863119878



119885 (119872)


119863119890119863119878(int


119866exp ( 119894

ℎ119878119864119867 [119890119872]))

(16)














4= 119863


1198774= 119860cork⋃

1198632


(17)






times 1198632rarr 119872 from 119891




0times119863

2 with1198632

1times119863

2 where1198632

0 119863

2

1sub





times 1198632 120597119863

2times 119863




1 120597


119899+1 120597

minus119879119899+1) is an


ℓ 120597




119899)cup120597

minus119879119899


CH = ⋃

ℓ=0

119879minus

ℓ (18)


119899997893rarr 119879

119899+1)


times R2 120597119863

2times R2



119899(119878

1times119863



2 in 1198991198781times 119878

2= 120597(

1198991198781times 119863




1198634 or the

decomposition

1198783times120579R = 119877

4 119863

4= (119860cork 119863

4)⋃

1198632

CH (19)





4


1198783times120579R = 119872(119878

3 120597119860cork)⋃

1198632

CH (20)




1198911= 119909

2+ 119910

2+ 119911

2minus 119905

2 for the 1-handle

1198912= 119909

2+ 119910

2minus 119911

2minus 119905



119894(119909 119910 119911 119905) = 119862 =


CH = ⋃


ℓ) cup 120597

minus119879ℓ) (22)




ℓ(and


1 119862) and 119871(119891

2 119862) is a






1015840sub 119878 with 1205971198631015840


1(119873) of











1198783times120579R

= 119872(1198783 120597119860cork)⋃

1198632

( ⋃


ℓ) cup 120597

minus119879ℓ))

(23)




120579R is



119885(1198783times120579R) = prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3)

times (prod

ℓ

119885 (int (119879ℓ))119885 (CHcork)

(24)


⟨1198783times120579R |O| 119878

3times120579R⟩ (25)


1rarr Σ


119885(1198783times120579R) = ⟨Σ

1

10038161003816100381610038161198791198721003816100381610038161003816 Σ2⟩ ⟨Σ2

10038161003816100381610038161198791198721003816100381610038161003816 Σ3⟩ sdot sdot sdot (26)









(1198783times120579R)

TOP= (119901 119905TOP) | 119901 isin 119878

3 119905TOP isin R


(27)





3timesR



sequence Σ1rarr Σ

2rarr Σ



2)cup

Σ2119872(Σ

2 Σ

3)cup

Σ3 of


1 Σ





119872(Σ1 Σ



3times120579R The




1 Σ


120579R




120579R Because


3 120597119860cork) between 119878

3


119860 (119890 119878) = int119878

1198892120590radic119864119886119864119887119899

119886119899119887 (28)



⟨1198783|119860 (119890 119878)| 120597119860cork⟩

=1

119885 (119872(1198783 120597119860cork))

times int119863119890119860 (119890 119878) exp ( 119894ℎ119878119864119867[119890119872 (119878

3 120597119860cork])

(29)


3 120597119860cork) is



surface

119878 = cupℓ119878ℓ (30)


119860ℓ(119890 119878

ℓ) = int

119878ℓ

1198892120590radic119864119886119864119887119899

119886119899119887 (31)

so that

⟨120597minus119879ℓ

1003816100381610038161003816119860ℓ1015840(119890 119878

ℓ1015840)1003816100381610038161003816 120597

minus119879ℓ+1⟩ = ⟨120597

minus119879ℓ

1003816100381610038161003816119860ℓ(119890 119878

ℓ)1003816100381610038161003816 120597

minus119879ℓ+1⟩ 120575

ℓℓ1015840

⟨1198783|119860 (119890 119878)| 120597119860cork⟩ = sum

ℓ

⟨120597minus119879ℓ

1003816100381610038161003816119860ℓ(119890 119878

ℓ)1003816100381610038161003816 120597

minus119879ℓ+1⟩

(32)


⟨120597minus1198790

10038161003816100381610038161198600(119890 119878

0)1003816100381610038161003816 120597

minus1198791⟩ = 119886

2

0 (33)






3-spheres Σ0 Σ


intΣ01

3

119877radicℎ1198893119909 = 119871 sdot CS (Σ

01) (34)


1denoted by


01) (defined



119871 sdot CS (Σ01 119860) = 119871

3sdotCS (Σ

01)

1198712= int

Σ01

CS (Σ01)

1198712radicℎ119889

3119909

(35)

by using

1198713= vol (Σ

01) = int

Σ01

radicℎ1198893119909 (36)

Together with

3119877 =

3119896

1198862 (37)


3119896

1198862=CS (Σ

01)

1198712 (38)


120599 =1198862

1198712=

3

CS (Σ01) (39)


1198891199042= 119889119905

2minus 119886(119905)

2ℎ119894119896119889119909

119894119889119909

119896 (40)


(119886

119886)

2

=1

1198712 (41)


1 Σ

2) between Σ

1and Σ

2we have two cases


1(say 119896(Σ

1) = +1)


2(say 119896(Σ



1 Σ




119905 997891997888rarr +1 0 le 119905



1 Σ

2) is compact)


1198891198862=1198862

1198712119889119905

2= 120599119889119905

2 (43)





119899+1at level 119899 + 1

1198891198862|119899+1

sim 120599 sdot 119889119886210038161003816100381610038161003816119899

(44)


2+ 119889119910

2)119910


2119910


119889119886210038161003816100381610038161003816119899+1

=120599

119899 + 1sdot 119889119886

210038161003816100381610038161003816119899=

120599119899+1

(119899 + 1)sdot 119889119886

2100381610038161003816100381610038160 (45)


1198862=

infin

sum

119899=0

(119889119886210038161003816100381610038161003816119899) = 119886

2

0sdot

infin

sum

119899=0

1

119899120599119899

= 1198862

0sdot exp (120599) = 1198862

0sdot 119897scale

(46)


with 1198891198862|0= 119886

2



⟨1198783|119860 (119890 119878)| 120597119860cork⟩

= ((prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3)

times

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩)

times(prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3))

minus1

)

=

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

(47)


⟨1198783|119860 (119890 119878)| 120597119860cork⟩ =

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

= 1198862

0sdot

infin

sum

119899=0

1

119899(

3

CS (120597119860cork))

119899

= 1198862

0sdot exp( 3

CS (120597119860cork))

(48)



1 From the




0and Σ



Σ1

Ψ

997888rarr R

120601 darr 119894119889

Σ0

120595

997888rarr R

(49)


0and Σ







1)


0to the



1)


minus10038171003817100381710038171206011003817100381710038171003817

2= minus (120601

2

1+ 120601

2

2+ 120601

2

3) 997888rarr minus119890

minus21206011 (1 + 1206012

2+ 120601

2

3) (50)


1 Σ


0rarr


1 Σ

1rarr R and


0 Σ

1) as


119871 = 119877 + (120597120583120601)

2

minus120588

2(1 minus exp (minus120582120601))2 (51)




119871 = 119877 +1

611987221198772 (52)





1198783 cork997888rarr 120597119860cork

tower997888rarr 119875119875 (53)



119886 = 1198860sdot exp( 3

2 sdot CS (120597119860cork)+

3

2 sdot CS (119875119875)) (54)


⟨1198783|119860 (119890 119878)| 119875119875⟩ = 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(55)


⟨1198783timesR |119860 (119890 119878)| 119878

3timesR⟩

= sum


3times120579R |119860 (119890 119878)| 119878

3times120579R⟩

= 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(56)




1198783997888rarr 120597119860cork (57)


1198860= 119871

119875= radic

ℎ119866

1198883 (58)


119886 = 119871119875sdot exp( 3

2 sdot CS(Σ (2 5 7)) (59)


119871119875sdot exp(140

3) asymp 75 sdot 10

minus15119898 (60)


1198783997888rarr Σ (2 5 7) 997888rarr 119875119875 (61)


119886 = 119871119875sdot exp(140

3+ 90) asymp 914 sdot 10

24119898 asymp 10

9119871119895 (62)





120599 =3

2 sdot CS (Σ (2 5 7)) (63)



(1 + 120599 + (12059922) + (12059936)) (64)



=1

(1 + 120599 + (12059922) + (12059936))asymp 55325 sdot 10

minus5

(65)







Λ =1

1198712

119875

sdot exp(minus 3


3

CS (119875119875)) (67)


Λ sdot 1198712



minus118 (68)



0(expressing the


1198712

119888=

1198882

31198672

0

(69)


ΩΛ=

1198885

3ℎ1198661198672

0

sdot exp(minus 3


3

CS (119875119875)) (70)


1198670= 74

km119904 sdotMpc

(71)


ΩΛ= 0729 (72)


(1198670)Planck = 68

km119904 sdotMpc

(73)

for which we obtain

ΩΛasymp 088 (74)


(ΩΛ)Planck = 0683 (75)


119878 =119860 sdot 120574

0

4 sdot 120574 sdot 1198712

119875

sdot 2120587 (76)

with

1205740=


(77)


119875


1

1198712

119875

997888rarr1

1198712

119875


=2120587120574

0

1198712

119875

asymp 080037 sdot1

1198712

119875

(78)


ΩΛ=

1198885

3ℎ1198661198672

0



3

CS (119875119875))

(79)


ΩΛasymp 0704 (80)


[52]

1205740=


(81)


ΩΛasymp 06836 (82)


7 Conclusion


Appendices








119865119860= 119889119860 + 119860 and 119860 (B1)


and Chern class

1198622=

1

81205872int119872

tr (119865119860and 119865

119860) (B2)


int119872

tr (119865119860and 119865

119860) = int

120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B3)


CS (120597119872119860) = 1

81205872int120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B4)





119896 =1

241205872int120597119872

(119892minus1119889119892)

3

isin Z (B6)


CS (120597119872119860) mod 1 (B7)


120575CS (120597119872119860) = 0 (B8)


119889119860 + 119860 and 119860 = 0 (B9)



120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (B10)


120591 (Σ) =1

81205872intΣtimesR

tr (119865119860and 119865

119860) (B11)


1

81205872intΣtimesR

tr (119865119860and 119865

119860)

10038161003816100381610038161003816100381610038161003816

le1

81205872intΣtimesR

tr (119865119860and lowast119865

119860) (B12)


119860 Thus the






119897

119902+119898

119903lt 1 119897 mod 2 = 119898 mod 2

119896

119901+119897

119902+119898

119903gt 1

119896

119901minus119897

119902+119898

119903lt 1

119896

119901+119897

119902minus119898

119903lt 1

(C1)


CS (120572) = 1198902


with




⟨1 1 1⟩CS = 1

120

⟨1 1 3⟩CS = 49

120

(C4)


⟨1 1 3⟩CS = 81

280

⟨1 3 1⟩CS = 9

280

⟨1 2 2⟩CS = 169280

⟨1 2 4⟩CS = 249280

(C5)


120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (C6)


120591 (119875) =1

120 120591 (Σ (2 5 7)) =

9

280 (C7)

And we are done




Acknowledgment


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119894119891= Σ

119894⊔ Σ

119891is the


119891to which




integral

⟨119860|Σ119891

10038161003816100381610038161198791198721003816100381610038161003816 119860|Σ119894

⟩

= int119860|120597119872119894119891

119863119860119863119890 exp( 119894ℎ119878119864119867[119890 119860119872

119894119891])

(2)



119894rarr H



119886





119892119886119887= 119890

119868

119886119890119869


119878119864119867[119890 119860119872

119894119891 120597119872

119894119891]

= int119872119894119891

120598119868119869KL (119890

119868and 119890

119869and (119889119860 + 119860 and 119860)

KL)

+ int120597119872119894119891

120598119868119869KL (119890

119868and 119890

119869and 119860

KL)

(3)


119868and

119890119869and119860



119894= Σ

119891are


119891and H = H

119894=




119885(1198721015840) = trH119879119872 = int119863119860119863119890 exp( 119894

ℎ119878119864119867 [119890 119860119872 120597119872])

(4)


119894119891 the


119872|119860|

Σ119894⟩ is the




⟨Σ119891

10038161003816100381610038161198791198721003816100381610038161003816 Σ119894⟩ = ⟨119860|Σ119891

10038161003816100381610038161198791198721003816100381610038161003816 119860|Σ119894

⟩ (5)


119894119891cupΣ119891119872

119891ℎ

with the amplitudes

⟨Σℎ

10038161003816100381610038161198791198721003816100381610038161003816 Σ119894⟩ = sum

119860|Σ119891

⟨Σℎ

10038161003816100381610038161198791198721003816100381610038161003816 Σ119891⟩ ⟨Σ119891

10038161003816100381610038161198791198721003816100381610038161003816 Σ119894⟩ (6)



119878120597[Σ

119891] = int

Σ119891

120598119868119869KL (119890

119868and 119890

119869and 119860

KL) = int

Σ119891

119867radicℎ1198893119909 (7)





rarr



119878plusmn




by 119878Σ= 120580

lowast119878+


lowastΦ for the


+

119872)) Let nabla119872

119883 nabla

Σ

119883be the


nabla119872

119883(Φ) = nabla

Σ

119883120601 minus

1

2(nabla

119883

rarr

119873) sdotrarr

119873 sdot120601 (8)




rarr


119883

rarr


119863119872Φ = 119863

3119863120601 minus 119867120601 (9)


119863119872Φ = 0 (10)


1198633119863120601 = 119867120601 (11)





1015840 Σ

1015840997893rarr 119872




119891 ∘ 119868 = 1198681015840∘ 119892 (12)


119892120590(Σ)


1(ΣZ




119892120590(Σ) is denoted by 119867

119892120590


119892120590to 119867

119891lowast119892119891lowast120590


119878120597[Σ

119891 ℎ] = int

Σ119891

119867radicℎ1198893119909 = int

Σ119891

120601 1198633119863120601119889

3119909 (13)




119885(Σ119891) = int119863ℎ exp( 119894

ℎ119878 [Σ

119891 ℎ])

= int119863120601119863120601 exp( 119894ℎintΣ119891

120601 1198633119863120601119889

3119909)


(14)






119878119864119867 [119890 119860119872] = int

119872

120598119868119869KL (119890

119868and 119890

119869and (119889119860 + 119860 and 119860)

KL) (15)




(ii) integration119863119890119863119878



119885 (119872)


119863119890119863119878(int


119866exp ( 119894

ℎ119878119864119867 [119890119872]))

(16)














4= 119863


1198774= 119860cork⋃

1198632


(17)






times 1198632rarr 119872 from 119891




0times119863

2 with1198632

1times119863

2 where1198632

0 119863

2

1sub





times 1198632 120597119863

2times 119863




1 120597


119899+1 120597

minus119879119899+1) is an


ℓ 120597




119899)cup120597

minus119879119899


CH = ⋃

ℓ=0

119879minus

ℓ (18)


119899997893rarr 119879

119899+1)


times R2 120597119863

2times R2



119899(119878

1times119863



2 in 1198991198781times 119878

2= 120597(

1198991198781times 119863




1198634 or the

decomposition

1198783times120579R = 119877

4 119863

4= (119860cork 119863

4)⋃

1198632

CH (19)





4


1198783times120579R = 119872(119878

3 120597119860cork)⋃

1198632

CH (20)




1198911= 119909

2+ 119910

2+ 119911

2minus 119905

2 for the 1-handle

1198912= 119909

2+ 119910

2minus 119911

2minus 119905



119894(119909 119910 119911 119905) = 119862 =


CH = ⋃


ℓ) cup 120597

minus119879ℓ) (22)




ℓ(and


1 119862) and 119871(119891

2 119862) is a






1015840sub 119878 with 1205971198631015840


1(119873) of











1198783times120579R

= 119872(1198783 120597119860cork)⋃

1198632

( ⋃


ℓ) cup 120597

minus119879ℓ))

(23)




120579R is



119885(1198783times120579R) = prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3)

times (prod

ℓ

119885 (int (119879ℓ))119885 (CHcork)

(24)


⟨1198783times120579R |O| 119878

3times120579R⟩ (25)


1rarr Σ


119885(1198783times120579R) = ⟨Σ

1

10038161003816100381610038161198791198721003816100381610038161003816 Σ2⟩ ⟨Σ2

10038161003816100381610038161198791198721003816100381610038161003816 Σ3⟩ sdot sdot sdot (26)









(1198783times120579R)

TOP= (119901 119905TOP) | 119901 isin 119878

3 119905TOP isin R


(27)





3timesR



sequence Σ1rarr Σ

2rarr Σ



2)cup

Σ2119872(Σ

2 Σ

3)cup

Σ3 of


1 Σ





119872(Σ1 Σ



3times120579R The




1 Σ


120579R




120579R Because


3 120597119860cork) between 119878

3


119860 (119890 119878) = int119878

1198892120590radic119864119886119864119887119899

119886119899119887 (28)



⟨1198783|119860 (119890 119878)| 120597119860cork⟩

=1

119885 (119872(1198783 120597119860cork))

times int119863119890119860 (119890 119878) exp ( 119894ℎ119878119864119867[119890119872 (119878

3 120597119860cork])

(29)


3 120597119860cork) is



surface

119878 = cupℓ119878ℓ (30)


119860ℓ(119890 119878

ℓ) = int

119878ℓ

1198892120590radic119864119886119864119887119899

119886119899119887 (31)

so that

⟨120597minus119879ℓ

1003816100381610038161003816119860ℓ1015840(119890 119878

ℓ1015840)1003816100381610038161003816 120597

minus119879ℓ+1⟩ = ⟨120597

minus119879ℓ

1003816100381610038161003816119860ℓ(119890 119878

ℓ)1003816100381610038161003816 120597

minus119879ℓ+1⟩ 120575

ℓℓ1015840

⟨1198783|119860 (119890 119878)| 120597119860cork⟩ = sum

ℓ

⟨120597minus119879ℓ

1003816100381610038161003816119860ℓ(119890 119878

ℓ)1003816100381610038161003816 120597

minus119879ℓ+1⟩

(32)


⟨120597minus1198790

10038161003816100381610038161198600(119890 119878

0)1003816100381610038161003816 120597

minus1198791⟩ = 119886

2

0 (33)






3-spheres Σ0 Σ


intΣ01

3

119877radicℎ1198893119909 = 119871 sdot CS (Σ

01) (34)


1denoted by


01) (defined



119871 sdot CS (Σ01 119860) = 119871

3sdotCS (Σ

01)

1198712= int

Σ01

CS (Σ01)

1198712radicℎ119889

3119909

(35)

by using

1198713= vol (Σ

01) = int

Σ01

radicℎ1198893119909 (36)

Together with

3119877 =

3119896

1198862 (37)


3119896

1198862=CS (Σ

01)

1198712 (38)


120599 =1198862

1198712=

3

CS (Σ01) (39)


1198891199042= 119889119905

2minus 119886(119905)

2ℎ119894119896119889119909

119894119889119909

119896 (40)


(119886

119886)

2

=1

1198712 (41)


1 Σ

2) between Σ

1and Σ

2we have two cases


1(say 119896(Σ

1) = +1)


2(say 119896(Σ



1 Σ




119905 997891997888rarr +1 0 le 119905



1 Σ

2) is compact)


1198891198862=1198862

1198712119889119905

2= 120599119889119905

2 (43)





119899+1at level 119899 + 1

1198891198862|119899+1

sim 120599 sdot 119889119886210038161003816100381610038161003816119899

(44)


2+ 119889119910

2)119910


2119910


119889119886210038161003816100381610038161003816119899+1

=120599

119899 + 1sdot 119889119886

210038161003816100381610038161003816119899=

120599119899+1

(119899 + 1)sdot 119889119886

2100381610038161003816100381610038160 (45)


1198862=

infin

sum

119899=0

(119889119886210038161003816100381610038161003816119899) = 119886

2

0sdot

infin

sum

119899=0

1

119899120599119899

= 1198862

0sdot exp (120599) = 1198862

0sdot 119897scale

(46)


with 1198891198862|0= 119886

2



⟨1198783|119860 (119890 119878)| 120597119860cork⟩

= ((prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3)

times

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩)

times(prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3))

minus1

)

=

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

(47)


⟨1198783|119860 (119890 119878)| 120597119860cork⟩ =

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

= 1198862

0sdot

infin

sum

119899=0

1

119899(

3

CS (120597119860cork))

119899

= 1198862

0sdot exp( 3

CS (120597119860cork))

(48)



1 From the




0and Σ



Σ1

Ψ

997888rarr R

120601 darr 119894119889

Σ0

120595

997888rarr R

(49)


0and Σ







1)


0to the



1)


minus10038171003817100381710038171206011003817100381710038171003817

2= minus (120601

2

1+ 120601

2

2+ 120601

2

3) 997888rarr minus119890

minus21206011 (1 + 1206012

2+ 120601

2

3) (50)


1 Σ


0rarr


1 Σ

1rarr R and


0 Σ

1) as


119871 = 119877 + (120597120583120601)

2

minus120588

2(1 minus exp (minus120582120601))2 (51)




119871 = 119877 +1

611987221198772 (52)





1198783 cork997888rarr 120597119860cork

tower997888rarr 119875119875 (53)



119886 = 1198860sdot exp( 3

2 sdot CS (120597119860cork)+

3

2 sdot CS (119875119875)) (54)


⟨1198783|119860 (119890 119878)| 119875119875⟩ = 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(55)


⟨1198783timesR |119860 (119890 119878)| 119878

3timesR⟩

= sum


3times120579R |119860 (119890 119878)| 119878

3times120579R⟩

= 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(56)




1198783997888rarr 120597119860cork (57)


1198860= 119871

119875= radic

ℎ119866

1198883 (58)


119886 = 119871119875sdot exp( 3

2 sdot CS(Σ (2 5 7)) (59)


119871119875sdot exp(140

3) asymp 75 sdot 10

minus15119898 (60)


1198783997888rarr Σ (2 5 7) 997888rarr 119875119875 (61)


119886 = 119871119875sdot exp(140

3+ 90) asymp 914 sdot 10

24119898 asymp 10

9119871119895 (62)





120599 =3

2 sdot CS (Σ (2 5 7)) (63)



(1 + 120599 + (12059922) + (12059936)) (64)



=1

(1 + 120599 + (12059922) + (12059936))asymp 55325 sdot 10

minus5

(65)







Λ =1

1198712

119875

sdot exp(minus 3


3

CS (119875119875)) (67)


Λ sdot 1198712



minus118 (68)



0(expressing the


1198712

119888=

1198882

31198672

0

(69)


ΩΛ=

1198885

3ℎ1198661198672

0

sdot exp(minus 3


3

CS (119875119875)) (70)


1198670= 74

km119904 sdotMpc

(71)


ΩΛ= 0729 (72)


(1198670)Planck = 68

km119904 sdotMpc

(73)

for which we obtain

ΩΛasymp 088 (74)


(ΩΛ)Planck = 0683 (75)


119878 =119860 sdot 120574

0

4 sdot 120574 sdot 1198712

119875

sdot 2120587 (76)

with

1205740=


(77)


119875


1

1198712

119875

997888rarr1

1198712

119875


=2120587120574

0

1198712

119875

asymp 080037 sdot1

1198712

119875

(78)


ΩΛ=

1198885

3ℎ1198661198672

0



3

CS (119875119875))

(79)


ΩΛasymp 0704 (80)


[52]

1205740=


(81)


ΩΛasymp 06836 (82)


7 Conclusion


Appendices








119865119860= 119889119860 + 119860 and 119860 (B1)


and Chern class

1198622=

1

81205872int119872

tr (119865119860and 119865

119860) (B2)


int119872

tr (119865119860and 119865

119860) = int

120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B3)


CS (120597119872119860) = 1

81205872int120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B4)





119896 =1

241205872int120597119872

(119892minus1119889119892)

3

isin Z (B6)


CS (120597119872119860) mod 1 (B7)


120575CS (120597119872119860) = 0 (B8)


119889119860 + 119860 and 119860 = 0 (B9)



120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (B10)


120591 (Σ) =1

81205872intΣtimesR

tr (119865119860and 119865

119860) (B11)


1

81205872intΣtimesR

tr (119865119860and 119865

119860)

10038161003816100381610038161003816100381610038161003816

le1

81205872intΣtimesR

tr (119865119860and lowast119865

119860) (B12)


119860 Thus the






119897

119902+119898

119903lt 1 119897 mod 2 = 119898 mod 2

119896

119901+119897

119902+119898

119903gt 1

119896

119901minus119897

119902+119898

119903lt 1

119896

119901+119897

119902minus119898

119903lt 1

(C1)


CS (120572) = 1198902


with




⟨1 1 1⟩CS = 1

120

⟨1 1 3⟩CS = 49

120

(C4)


⟨1 1 3⟩CS = 81

280

⟨1 3 1⟩CS = 9

280

⟨1 2 2⟩CS = 169280

⟨1 2 4⟩CS = 249280

(C5)


120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (C6)


120591 (119875) =1

120 120591 (Σ (2 5 7)) =

9

280 (C7)

And we are done




Acknowledgment


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






1015840 Σ

1015840997893rarr 119872




119891 ∘ 119868 = 1198681015840∘ 119892 (12)


119892120590(Σ)


1(ΣZ




119892120590(Σ) is denoted by 119867

119892120590


119892120590to 119867

119891lowast119892119891lowast120590


119878120597[Σ

119891 ℎ] = int

Σ119891

119867radicℎ1198893119909 = int

Σ119891

120601 1198633119863120601119889

3119909 (13)




119885(Σ119891) = int119863ℎ exp( 119894

ℎ119878 [Σ

119891 ℎ])

= int119863120601119863120601 exp( 119894ℎintΣ119891

120601 1198633119863120601119889

3119909)


(14)






119878119864119867 [119890 119860119872] = int

119872

120598119868119869KL (119890

119868and 119890

119869and (119889119860 + 119860 and 119860)

KL) (15)




(ii) integration119863119890119863119878



119885 (119872)


119863119890119863119878(int


119866exp ( 119894

ℎ119878119864119867 [119890119872]))

(16)














4= 119863


1198774= 119860cork⋃

1198632


(17)






times 1198632rarr 119872 from 119891




0times119863

2 with1198632

1times119863

2 where1198632

0 119863

2

1sub





times 1198632 120597119863

2times 119863




1 120597


119899+1 120597

minus119879119899+1) is an


ℓ 120597




119899)cup120597

minus119879119899


CH = ⋃

ℓ=0

119879minus

ℓ (18)


119899997893rarr 119879

119899+1)


times R2 120597119863

2times R2



119899(119878

1times119863



2 in 1198991198781times 119878

2= 120597(

1198991198781times 119863




1198634 or the

decomposition

1198783times120579R = 119877

4 119863

4= (119860cork 119863

4)⋃

1198632

CH (19)





4


1198783times120579R = 119872(119878

3 120597119860cork)⋃

1198632

CH (20)




1198911= 119909

2+ 119910

2+ 119911

2minus 119905

2 for the 1-handle

1198912= 119909

2+ 119910

2minus 119911

2minus 119905



119894(119909 119910 119911 119905) = 119862 =


CH = ⋃


ℓ) cup 120597

minus119879ℓ) (22)




ℓ(and


1 119862) and 119871(119891

2 119862) is a






1015840sub 119878 with 1205971198631015840


1(119873) of











1198783times120579R

= 119872(1198783 120597119860cork)⋃

1198632

( ⋃


ℓ) cup 120597

minus119879ℓ))

(23)




120579R is



119885(1198783times120579R) = prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3)

times (prod

ℓ

119885 (int (119879ℓ))119885 (CHcork)

(24)


⟨1198783times120579R |O| 119878

3times120579R⟩ (25)


1rarr Σ


119885(1198783times120579R) = ⟨Σ

1

10038161003816100381610038161198791198721003816100381610038161003816 Σ2⟩ ⟨Σ2

10038161003816100381610038161198791198721003816100381610038161003816 Σ3⟩ sdot sdot sdot (26)









(1198783times120579R)

TOP= (119901 119905TOP) | 119901 isin 119878

3 119905TOP isin R


(27)





3timesR



sequence Σ1rarr Σ

2rarr Σ



2)cup

Σ2119872(Σ

2 Σ

3)cup

Σ3 of


1 Σ





119872(Σ1 Σ



3times120579R The




1 Σ


120579R




120579R Because


3 120597119860cork) between 119878

3


119860 (119890 119878) = int119878

1198892120590radic119864119886119864119887119899

119886119899119887 (28)



⟨1198783|119860 (119890 119878)| 120597119860cork⟩

=1

119885 (119872(1198783 120597119860cork))

times int119863119890119860 (119890 119878) exp ( 119894ℎ119878119864119867[119890119872 (119878

3 120597119860cork])

(29)


3 120597119860cork) is



surface

119878 = cupℓ119878ℓ (30)


119860ℓ(119890 119878

ℓ) = int

119878ℓ

1198892120590radic119864119886119864119887119899

119886119899119887 (31)

so that

⟨120597minus119879ℓ

1003816100381610038161003816119860ℓ1015840(119890 119878

ℓ1015840)1003816100381610038161003816 120597

minus119879ℓ+1⟩ = ⟨120597

minus119879ℓ

1003816100381610038161003816119860ℓ(119890 119878

ℓ)1003816100381610038161003816 120597

minus119879ℓ+1⟩ 120575

ℓℓ1015840

⟨1198783|119860 (119890 119878)| 120597119860cork⟩ = sum

ℓ

⟨120597minus119879ℓ

1003816100381610038161003816119860ℓ(119890 119878

ℓ)1003816100381610038161003816 120597

minus119879ℓ+1⟩

(32)


⟨120597minus1198790

10038161003816100381610038161198600(119890 119878

0)1003816100381610038161003816 120597

minus1198791⟩ = 119886

2

0 (33)






3-spheres Σ0 Σ


intΣ01

3

119877radicℎ1198893119909 = 119871 sdot CS (Σ

01) (34)


1denoted by


01) (defined



119871 sdot CS (Σ01 119860) = 119871

3sdotCS (Σ

01)

1198712= int

Σ01

CS (Σ01)

1198712radicℎ119889

3119909

(35)

by using

1198713= vol (Σ

01) = int

Σ01

radicℎ1198893119909 (36)

Together with

3119877 =

3119896

1198862 (37)


3119896

1198862=CS (Σ

01)

1198712 (38)


120599 =1198862

1198712=

3

CS (Σ01) (39)


1198891199042= 119889119905

2minus 119886(119905)

2ℎ119894119896119889119909

119894119889119909

119896 (40)


(119886

119886)

2

=1

1198712 (41)


1 Σ

2) between Σ

1and Σ

2we have two cases


1(say 119896(Σ

1) = +1)


2(say 119896(Σ



1 Σ




119905 997891997888rarr +1 0 le 119905



1 Σ

2) is compact)


1198891198862=1198862

1198712119889119905

2= 120599119889119905

2 (43)





119899+1at level 119899 + 1

1198891198862|119899+1

sim 120599 sdot 119889119886210038161003816100381610038161003816119899

(44)


2+ 119889119910

2)119910


2119910


119889119886210038161003816100381610038161003816119899+1

=120599

119899 + 1sdot 119889119886

210038161003816100381610038161003816119899=

120599119899+1

(119899 + 1)sdot 119889119886

2100381610038161003816100381610038160 (45)


1198862=

infin

sum

119899=0

(119889119886210038161003816100381610038161003816119899) = 119886

2

0sdot

infin

sum

119899=0

1

119899120599119899

= 1198862

0sdot exp (120599) = 1198862

0sdot 119897scale

(46)


with 1198891198862|0= 119886

2



⟨1198783|119860 (119890 119878)| 120597119860cork⟩

= ((prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3)

times

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩)

times(prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3))

minus1

)

=

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

(47)


⟨1198783|119860 (119890 119878)| 120597119860cork⟩ =

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

= 1198862

0sdot

infin

sum

119899=0

1

119899(

3

CS (120597119860cork))

119899

= 1198862

0sdot exp( 3

CS (120597119860cork))

(48)



1 From the




0and Σ



Σ1

Ψ

997888rarr R

120601 darr 119894119889

Σ0

120595

997888rarr R

(49)


0and Σ







1)


0to the



1)


minus10038171003817100381710038171206011003817100381710038171003817

2= minus (120601

2

1+ 120601

2

2+ 120601

2

3) 997888rarr minus119890

minus21206011 (1 + 1206012

2+ 120601

2

3) (50)


1 Σ


0rarr


1 Σ

1rarr R and


0 Σ

1) as


119871 = 119877 + (120597120583120601)

2

minus120588

2(1 minus exp (minus120582120601))2 (51)




119871 = 119877 +1

611987221198772 (52)





1198783 cork997888rarr 120597119860cork

tower997888rarr 119875119875 (53)



119886 = 1198860sdot exp( 3

2 sdot CS (120597119860cork)+

3

2 sdot CS (119875119875)) (54)


⟨1198783|119860 (119890 119878)| 119875119875⟩ = 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(55)


⟨1198783timesR |119860 (119890 119878)| 119878

3timesR⟩

= sum


3times120579R |119860 (119890 119878)| 119878

3times120579R⟩

= 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(56)




1198783997888rarr 120597119860cork (57)


1198860= 119871

119875= radic

ℎ119866

1198883 (58)


119886 = 119871119875sdot exp( 3

2 sdot CS(Σ (2 5 7)) (59)


119871119875sdot exp(140

3) asymp 75 sdot 10

minus15119898 (60)


1198783997888rarr Σ (2 5 7) 997888rarr 119875119875 (61)


119886 = 119871119875sdot exp(140

3+ 90) asymp 914 sdot 10

24119898 asymp 10

9119871119895 (62)





120599 =3

2 sdot CS (Σ (2 5 7)) (63)



(1 + 120599 + (12059922) + (12059936)) (64)



=1

(1 + 120599 + (12059922) + (12059936))asymp 55325 sdot 10

minus5

(65)







Λ =1

1198712

119875

sdot exp(minus 3


3

CS (119875119875)) (67)


Λ sdot 1198712



minus118 (68)



0(expressing the


1198712

119888=

1198882

31198672

0

(69)


ΩΛ=

1198885

3ℎ1198661198672

0

sdot exp(minus 3


3

CS (119875119875)) (70)


1198670= 74

km119904 sdotMpc

(71)


ΩΛ= 0729 (72)


(1198670)Planck = 68

km119904 sdotMpc

(73)

for which we obtain

ΩΛasymp 088 (74)


(ΩΛ)Planck = 0683 (75)


119878 =119860 sdot 120574

0

4 sdot 120574 sdot 1198712

119875

sdot 2120587 (76)

with

1205740=


(77)


119875


1

1198712

119875

997888rarr1

1198712

119875


=2120587120574

0

1198712

119875

asymp 080037 sdot1

1198712

119875

(78)


ΩΛ=

1198885

3ℎ1198661198672

0



3

CS (119875119875))

(79)


ΩΛasymp 0704 (80)


[52]

1205740=


(81)


ΩΛasymp 06836 (82)


7 Conclusion


Appendices








119865119860= 119889119860 + 119860 and 119860 (B1)


and Chern class

1198622=

1

81205872int119872

tr (119865119860and 119865

119860) (B2)


int119872

tr (119865119860and 119865

119860) = int

120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B3)


CS (120597119872119860) = 1

81205872int120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B4)





119896 =1

241205872int120597119872

(119892minus1119889119892)

3

isin Z (B6)


CS (120597119872119860) mod 1 (B7)


120575CS (120597119872119860) = 0 (B8)


119889119860 + 119860 and 119860 = 0 (B9)



120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (B10)


120591 (Σ) =1

81205872intΣtimesR

tr (119865119860and 119865

119860) (B11)


1

81205872intΣtimesR

tr (119865119860and 119865

119860)

10038161003816100381610038161003816100381610038161003816

le1

81205872intΣtimesR

tr (119865119860and lowast119865

119860) (B12)


119860 Thus the






119897

119902+119898

119903lt 1 119897 mod 2 = 119898 mod 2

119896

119901+119897

119902+119898

119903gt 1

119896

119901minus119897

119902+119898

119903lt 1

119896

119901+119897

119902minus119898

119903lt 1

(C1)


CS (120572) = 1198902


with




⟨1 1 1⟩CS = 1

120

⟨1 1 3⟩CS = 49

120

(C4)


⟨1 1 3⟩CS = 81

280

⟨1 3 1⟩CS = 9

280

⟨1 2 2⟩CS = 169280

⟨1 2 4⟩CS = 249280

(C5)


120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (C6)


120591 (119875) =1

120 120591 (Σ (2 5 7)) =

9

280 (C7)

And we are done




Acknowledgment


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










4= 119863


1198774= 119860cork⋃

1198632


(17)






times 1198632rarr 119872 from 119891




0times119863

2 with1198632

1times119863

2 where1198632

0 119863

2

1sub





times 1198632 120597119863

2times 119863




1 120597


119899+1 120597

minus119879119899+1) is an


ℓ 120597




119899)cup120597

minus119879119899


CH = ⋃

ℓ=0

119879minus

ℓ (18)


119899997893rarr 119879

119899+1)


times R2 120597119863

2times R2



119899(119878

1times119863



2 in 1198991198781times 119878

2= 120597(

1198991198781times 119863




1198634 or the

decomposition

1198783times120579R = 119877

4 119863

4= (119860cork 119863

4)⋃

1198632

CH (19)





4


1198783times120579R = 119872(119878

3 120597119860cork)⋃

1198632

CH (20)




1198911= 119909

2+ 119910

2+ 119911

2minus 119905

2 for the 1-handle

1198912= 119909

2+ 119910

2minus 119911

2minus 119905



119894(119909 119910 119911 119905) = 119862 =


CH = ⋃


ℓ) cup 120597

minus119879ℓ) (22)




ℓ(and


1 119862) and 119871(119891

2 119862) is a






1015840sub 119878 with 1205971198631015840


1(119873) of











1198783times120579R

= 119872(1198783 120597119860cork)⋃

1198632

( ⋃


ℓ) cup 120597

minus119879ℓ))

(23)




120579R is



119885(1198783times120579R) = prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3)

times (prod

ℓ

119885 (int (119879ℓ))119885 (CHcork)

(24)


⟨1198783times120579R |O| 119878

3times120579R⟩ (25)


1rarr Σ


119885(1198783times120579R) = ⟨Σ

1

10038161003816100381610038161198791198721003816100381610038161003816 Σ2⟩ ⟨Σ2

10038161003816100381610038161198791198721003816100381610038161003816 Σ3⟩ sdot sdot sdot (26)









(1198783times120579R)

TOP= (119901 119905TOP) | 119901 isin 119878

3 119905TOP isin R


(27)





3timesR



sequence Σ1rarr Σ

2rarr Σ



2)cup

Σ2119872(Σ

2 Σ

3)cup

Σ3 of


1 Σ





119872(Σ1 Σ



3times120579R The




1 Σ


120579R




120579R Because


3 120597119860cork) between 119878

3


119860 (119890 119878) = int119878

1198892120590radic119864119886119864119887119899

119886119899119887 (28)



⟨1198783|119860 (119890 119878)| 120597119860cork⟩

=1

119885 (119872(1198783 120597119860cork))

times int119863119890119860 (119890 119878) exp ( 119894ℎ119878119864119867[119890119872 (119878

3 120597119860cork])

(29)


3 120597119860cork) is



surface

119878 = cupℓ119878ℓ (30)


119860ℓ(119890 119878

ℓ) = int

119878ℓ

1198892120590radic119864119886119864119887119899

119886119899119887 (31)

so that

⟨120597minus119879ℓ

1003816100381610038161003816119860ℓ1015840(119890 119878

ℓ1015840)1003816100381610038161003816 120597

minus119879ℓ+1⟩ = ⟨120597

minus119879ℓ

1003816100381610038161003816119860ℓ(119890 119878

ℓ)1003816100381610038161003816 120597

minus119879ℓ+1⟩ 120575

ℓℓ1015840

⟨1198783|119860 (119890 119878)| 120597119860cork⟩ = sum

ℓ

⟨120597minus119879ℓ

1003816100381610038161003816119860ℓ(119890 119878

ℓ)1003816100381610038161003816 120597

minus119879ℓ+1⟩

(32)


⟨120597minus1198790

10038161003816100381610038161198600(119890 119878

0)1003816100381610038161003816 120597

minus1198791⟩ = 119886

2

0 (33)






3-spheres Σ0 Σ


intΣ01

3

119877radicℎ1198893119909 = 119871 sdot CS (Σ

01) (34)


1denoted by


01) (defined



119871 sdot CS (Σ01 119860) = 119871

3sdotCS (Σ

01)

1198712= int

Σ01

CS (Σ01)

1198712radicℎ119889

3119909

(35)

by using

1198713= vol (Σ

01) = int

Σ01

radicℎ1198893119909 (36)

Together with

3119877 =

3119896

1198862 (37)


3119896

1198862=CS (Σ

01)

1198712 (38)


120599 =1198862

1198712=

3

CS (Σ01) (39)


1198891199042= 119889119905

2minus 119886(119905)

2ℎ119894119896119889119909

119894119889119909

119896 (40)


(119886

119886)

2

=1

1198712 (41)


1 Σ

2) between Σ

1and Σ

2we have two cases


1(say 119896(Σ

1) = +1)


2(say 119896(Σ



1 Σ




119905 997891997888rarr +1 0 le 119905



1 Σ

2) is compact)


1198891198862=1198862

1198712119889119905

2= 120599119889119905

2 (43)





119899+1at level 119899 + 1

1198891198862|119899+1

sim 120599 sdot 119889119886210038161003816100381610038161003816119899

(44)


2+ 119889119910

2)119910


2119910


119889119886210038161003816100381610038161003816119899+1

=120599

119899 + 1sdot 119889119886

210038161003816100381610038161003816119899=

120599119899+1

(119899 + 1)sdot 119889119886

2100381610038161003816100381610038160 (45)


1198862=

infin

sum

119899=0

(119889119886210038161003816100381610038161003816119899) = 119886

2

0sdot

infin

sum

119899=0

1

119899120599119899

= 1198862

0sdot exp (120599) = 1198862

0sdot 119897scale

(46)


with 1198891198862|0= 119886

2



⟨1198783|119860 (119890 119878)| 120597119860cork⟩

= ((prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3)

times

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩)

times(prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3))

minus1

)

=

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

(47)


⟨1198783|119860 (119890 119878)| 120597119860cork⟩ =

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

= 1198862

0sdot

infin

sum

119899=0

1

119899(

3

CS (120597119860cork))

119899

= 1198862

0sdot exp( 3

CS (120597119860cork))

(48)



1 From the




0and Σ



Σ1

Ψ

997888rarr R

120601 darr 119894119889

Σ0

120595

997888rarr R

(49)


0and Σ







1)


0to the



1)


minus10038171003817100381710038171206011003817100381710038171003817

2= minus (120601

2

1+ 120601

2

2+ 120601

2

3) 997888rarr minus119890

minus21206011 (1 + 1206012

2+ 120601

2

3) (50)


1 Σ


0rarr


1 Σ

1rarr R and


0 Σ

1) as


119871 = 119877 + (120597120583120601)

2

minus120588

2(1 minus exp (minus120582120601))2 (51)




119871 = 119877 +1

611987221198772 (52)





1198783 cork997888rarr 120597119860cork

tower997888rarr 119875119875 (53)



119886 = 1198860sdot exp( 3

2 sdot CS (120597119860cork)+

3

2 sdot CS (119875119875)) (54)


⟨1198783|119860 (119890 119878)| 119875119875⟩ = 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(55)


⟨1198783timesR |119860 (119890 119878)| 119878

3timesR⟩

= sum


3times120579R |119860 (119890 119878)| 119878

3times120579R⟩

= 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(56)




1198783997888rarr 120597119860cork (57)


1198860= 119871

119875= radic

ℎ119866

1198883 (58)


119886 = 119871119875sdot exp( 3

2 sdot CS(Σ (2 5 7)) (59)


119871119875sdot exp(140

3) asymp 75 sdot 10

minus15119898 (60)


1198783997888rarr Σ (2 5 7) 997888rarr 119875119875 (61)


119886 = 119871119875sdot exp(140

3+ 90) asymp 914 sdot 10

24119898 asymp 10

9119871119895 (62)





120599 =3

2 sdot CS (Σ (2 5 7)) (63)



(1 + 120599 + (12059922) + (12059936)) (64)



=1

(1 + 120599 + (12059922) + (12059936))asymp 55325 sdot 10

minus5

(65)







Λ =1

1198712

119875

sdot exp(minus 3


3

CS (119875119875)) (67)


Λ sdot 1198712



minus118 (68)



0(expressing the


1198712

119888=

1198882

31198672

0

(69)


ΩΛ=

1198885

3ℎ1198661198672

0

sdot exp(minus 3


3

CS (119875119875)) (70)


1198670= 74

km119904 sdotMpc

(71)


ΩΛ= 0729 (72)


(1198670)Planck = 68

km119904 sdotMpc

(73)

for which we obtain

ΩΛasymp 088 (74)


(ΩΛ)Planck = 0683 (75)


119878 =119860 sdot 120574

0

4 sdot 120574 sdot 1198712

119875

sdot 2120587 (76)

with

1205740=


(77)


119875


1

1198712

119875

997888rarr1

1198712

119875


=2120587120574

0

1198712

119875

asymp 080037 sdot1

1198712

119875

(78)


ΩΛ=

1198885

3ℎ1198661198672

0



3

CS (119875119875))

(79)


ΩΛasymp 0704 (80)


[52]

1205740=


(81)


ΩΛasymp 06836 (82)


7 Conclusion


Appendices








119865119860= 119889119860 + 119860 and 119860 (B1)


and Chern class

1198622=

1

81205872int119872

tr (119865119860and 119865

119860) (B2)


int119872

tr (119865119860and 119865

119860) = int

120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B3)


CS (120597119872119860) = 1

81205872int120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B4)





119896 =1

241205872int120597119872

(119892minus1119889119892)

3

isin Z (B6)


CS (120597119872119860) mod 1 (B7)


120575CS (120597119872119860) = 0 (B8)


119889119860 + 119860 and 119860 = 0 (B9)



120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (B10)


120591 (Σ) =1

81205872intΣtimesR

tr (119865119860and 119865

119860) (B11)


1

81205872intΣtimesR

tr (119865119860and 119865

119860)

10038161003816100381610038161003816100381610038161003816

le1

81205872intΣtimesR

tr (119865119860and lowast119865

119860) (B12)


119860 Thus the






119897

119902+119898

119903lt 1 119897 mod 2 = 119898 mod 2

119896

119901+119897

119902+119898

119903gt 1

119896

119901minus119897

119902+119898

119903lt 1

119896

119901+119897

119902minus119898

119903lt 1

(C1)


CS (120572) = 1198902


with




⟨1 1 1⟩CS = 1

120

⟨1 1 3⟩CS = 49

120

(C4)


⟨1 1 3⟩CS = 81

280

⟨1 3 1⟩CS = 9

280

⟨1 2 2⟩CS = 169280

⟨1 2 4⟩CS = 249280

(C5)


120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (C6)


120591 (119875) =1

120 120591 (Σ (2 5 7)) =

9

280 (C7)

And we are done




Acknowledgment


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1198911= 119909

2+ 119910

2+ 119911

2minus 119905

2 for the 1-handle

1198912= 119909

2+ 119910

2minus 119911

2minus 119905



119894(119909 119910 119911 119905) = 119862 =


CH = ⋃


ℓ) cup 120597

minus119879ℓ) (22)




ℓ(and


1 119862) and 119871(119891

2 119862) is a






1015840sub 119878 with 1205971198631015840


1(119873) of











1198783times120579R

= 119872(1198783 120597119860cork)⋃

1198632

( ⋃


ℓ) cup 120597

minus119879ℓ))

(23)




120579R is



119885(1198783times120579R) = prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3)

times (prod

ℓ

119885 (int (119879ℓ))119885 (CHcork)

(24)


⟨1198783times120579R |O| 119878

3times120579R⟩ (25)


1rarr Σ


119885(1198783times120579R) = ⟨Σ

1

10038161003816100381610038161198791198721003816100381610038161003816 Σ2⟩ ⟨Σ2

10038161003816100381610038161198791198721003816100381610038161003816 Σ3⟩ sdot sdot sdot (26)









(1198783times120579R)

TOP= (119901 119905TOP) | 119901 isin 119878

3 119905TOP isin R


(27)





3timesR



sequence Σ1rarr Σ

2rarr Σ



2)cup

Σ2119872(Σ

2 Σ

3)cup

Σ3 of


1 Σ





119872(Σ1 Σ



3times120579R The




1 Σ


120579R




120579R Because


3 120597119860cork) between 119878

3


119860 (119890 119878) = int119878

1198892120590radic119864119886119864119887119899

119886119899119887 (28)



⟨1198783|119860 (119890 119878)| 120597119860cork⟩

=1

119885 (119872(1198783 120597119860cork))

times int119863119890119860 (119890 119878) exp ( 119894ℎ119878119864119867[119890119872 (119878

3 120597119860cork])

(29)


3 120597119860cork) is



surface

119878 = cupℓ119878ℓ (30)


119860ℓ(119890 119878

ℓ) = int

119878ℓ

1198892120590radic119864119886119864119887119899

119886119899119887 (31)

so that

⟨120597minus119879ℓ

1003816100381610038161003816119860ℓ1015840(119890 119878

ℓ1015840)1003816100381610038161003816 120597

minus119879ℓ+1⟩ = ⟨120597

minus119879ℓ

1003816100381610038161003816119860ℓ(119890 119878

ℓ)1003816100381610038161003816 120597

minus119879ℓ+1⟩ 120575

ℓℓ1015840

⟨1198783|119860 (119890 119878)| 120597119860cork⟩ = sum

ℓ

⟨120597minus119879ℓ

1003816100381610038161003816119860ℓ(119890 119878

ℓ)1003816100381610038161003816 120597

minus119879ℓ+1⟩

(32)


⟨120597minus1198790

10038161003816100381610038161198600(119890 119878

0)1003816100381610038161003816 120597

minus1198791⟩ = 119886

2

0 (33)






3-spheres Σ0 Σ


intΣ01

3

119877radicℎ1198893119909 = 119871 sdot CS (Σ

01) (34)


1denoted by


01) (defined



119871 sdot CS (Σ01 119860) = 119871

3sdotCS (Σ

01)

1198712= int

Σ01

CS (Σ01)

1198712radicℎ119889

3119909

(35)

by using

1198713= vol (Σ

01) = int

Σ01

radicℎ1198893119909 (36)

Together with

3119877 =

3119896

1198862 (37)


3119896

1198862=CS (Σ

01)

1198712 (38)


120599 =1198862

1198712=

3

CS (Σ01) (39)


1198891199042= 119889119905

2minus 119886(119905)

2ℎ119894119896119889119909

119894119889119909

119896 (40)


(119886

119886)

2

=1

1198712 (41)


1 Σ

2) between Σ

1and Σ

2we have two cases


1(say 119896(Σ

1) = +1)


2(say 119896(Σ



1 Σ




119905 997891997888rarr +1 0 le 119905



1 Σ

2) is compact)


1198891198862=1198862

1198712119889119905

2= 120599119889119905

2 (43)





119899+1at level 119899 + 1

1198891198862|119899+1

sim 120599 sdot 119889119886210038161003816100381610038161003816119899

(44)


2+ 119889119910

2)119910


2119910


119889119886210038161003816100381610038161003816119899+1

=120599

119899 + 1sdot 119889119886

210038161003816100381610038161003816119899=

120599119899+1

(119899 + 1)sdot 119889119886

2100381610038161003816100381610038160 (45)


1198862=

infin

sum

119899=0

(119889119886210038161003816100381610038161003816119899) = 119886

2

0sdot

infin

sum

119899=0

1

119899120599119899

= 1198862

0sdot exp (120599) = 1198862

0sdot 119897scale

(46)


with 1198891198862|0= 119886

2



⟨1198783|119860 (119890 119878)| 120597119860cork⟩

= ((prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3)

times

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩)

times(prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3))

minus1

)

=

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

(47)


⟨1198783|119860 (119890 119878)| 120597119860cork⟩ =

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

= 1198862

0sdot

infin

sum

119899=0

1

119899(

3

CS (120597119860cork))

119899

= 1198862

0sdot exp( 3

CS (120597119860cork))

(48)



1 From the




0and Σ



Σ1

Ψ

997888rarr R

120601 darr 119894119889

Σ0

120595

997888rarr R

(49)


0and Σ







1)


0to the



1)


minus10038171003817100381710038171206011003817100381710038171003817

2= minus (120601

2

1+ 120601

2

2+ 120601

2

3) 997888rarr minus119890

minus21206011 (1 + 1206012

2+ 120601

2

3) (50)


1 Σ


0rarr


1 Σ

1rarr R and


0 Σ

1) as


119871 = 119877 + (120597120583120601)

2

minus120588

2(1 minus exp (minus120582120601))2 (51)




119871 = 119877 +1

611987221198772 (52)





1198783 cork997888rarr 120597119860cork

tower997888rarr 119875119875 (53)



119886 = 1198860sdot exp( 3

2 sdot CS (120597119860cork)+

3

2 sdot CS (119875119875)) (54)


⟨1198783|119860 (119890 119878)| 119875119875⟩ = 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(55)


⟨1198783timesR |119860 (119890 119878)| 119878

3timesR⟩

= sum


3times120579R |119860 (119890 119878)| 119878

3times120579R⟩

= 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(56)




1198783997888rarr 120597119860cork (57)


1198860= 119871

119875= radic

ℎ119866

1198883 (58)


119886 = 119871119875sdot exp( 3

2 sdot CS(Σ (2 5 7)) (59)


119871119875sdot exp(140

3) asymp 75 sdot 10

minus15119898 (60)


1198783997888rarr Σ (2 5 7) 997888rarr 119875119875 (61)


119886 = 119871119875sdot exp(140

3+ 90) asymp 914 sdot 10

24119898 asymp 10

9119871119895 (62)





120599 =3

2 sdot CS (Σ (2 5 7)) (63)



(1 + 120599 + (12059922) + (12059936)) (64)



=1

(1 + 120599 + (12059922) + (12059936))asymp 55325 sdot 10

minus5

(65)







Λ =1

1198712

119875

sdot exp(minus 3


3

CS (119875119875)) (67)


Λ sdot 1198712



minus118 (68)



0(expressing the


1198712

119888=

1198882

31198672

0

(69)


ΩΛ=

1198885

3ℎ1198661198672

0

sdot exp(minus 3


3

CS (119875119875)) (70)


1198670= 74

km119904 sdotMpc

(71)


ΩΛ= 0729 (72)


(1198670)Planck = 68

km119904 sdotMpc

(73)

for which we obtain

ΩΛasymp 088 (74)


(ΩΛ)Planck = 0683 (75)


119878 =119860 sdot 120574

0

4 sdot 120574 sdot 1198712

119875

sdot 2120587 (76)

with

1205740=


(77)


119875


1

1198712

119875

997888rarr1

1198712

119875


=2120587120574

0

1198712

119875

asymp 080037 sdot1

1198712

119875

(78)


ΩΛ=

1198885

3ℎ1198661198672

0



3

CS (119875119875))

(79)


ΩΛasymp 0704 (80)


[52]

1205740=


(81)


ΩΛasymp 06836 (82)


7 Conclusion


Appendices








119865119860= 119889119860 + 119860 and 119860 (B1)


and Chern class

1198622=

1

81205872int119872

tr (119865119860and 119865

119860) (B2)


int119872

tr (119865119860and 119865

119860) = int

120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B3)


CS (120597119872119860) = 1

81205872int120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B4)





119896 =1

241205872int120597119872

(119892minus1119889119892)

3

isin Z (B6)


CS (120597119872119860) mod 1 (B7)


120575CS (120597119872119860) = 0 (B8)


119889119860 + 119860 and 119860 = 0 (B9)



120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (B10)


120591 (Σ) =1

81205872intΣtimesR

tr (119865119860and 119865

119860) (B11)


1

81205872intΣtimesR

tr (119865119860and 119865

119860)

10038161003816100381610038161003816100381610038161003816

le1

81205872intΣtimesR

tr (119865119860and lowast119865

119860) (B12)


119860 Thus the






119897

119902+119898

119903lt 1 119897 mod 2 = 119898 mod 2

119896

119901+119897

119902+119898

119903gt 1

119896

119901minus119897

119902+119898

119903lt 1

119896

119901+119897

119902minus119898

119903lt 1

(C1)


CS (120572) = 1198902


with




⟨1 1 1⟩CS = 1

120

⟨1 1 3⟩CS = 49

120

(C4)


⟨1 1 3⟩CS = 81

280

⟨1 3 1⟩CS = 9

280

⟨1 2 2⟩CS = 169280

⟨1 2 4⟩CS = 249280

(C5)


120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (C6)


120591 (119875) =1

120 120591 (Σ (2 5 7)) =

9

280 (C7)

And we are done




Acknowledgment


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








(1198783times120579R)

TOP= (119901 119905TOP) | 119901 isin 119878

3 119905TOP isin R


(27)





3timesR



sequence Σ1rarr Σ

2rarr Σ



2)cup

Σ2119872(Σ

2 Σ

3)cup

Σ3 of


1 Σ





119872(Σ1 Σ



3times120579R The




1 Σ


120579R




120579R Because


3 120597119860cork) between 119878

3


119860 (119890 119878) = int119878

1198892120590radic119864119886119864119887119899

119886119899119887 (28)



⟨1198783|119860 (119890 119878)| 120597119860cork⟩

=1

119885 (119872(1198783 120597119860cork))

times int119863119890119860 (119890 119878) exp ( 119894ℎ119878119864119867[119890119872 (119878

3 120597119860cork])

(29)


3 120597119860cork) is



surface

119878 = cupℓ119878ℓ (30)


119860ℓ(119890 119878

ℓ) = int

119878ℓ

1198892120590radic119864119886119864119887119899

119886119899119887 (31)

so that

⟨120597minus119879ℓ

1003816100381610038161003816119860ℓ1015840(119890 119878

ℓ1015840)1003816100381610038161003816 120597

minus119879ℓ+1⟩ = ⟨120597

minus119879ℓ

1003816100381610038161003816119860ℓ(119890 119878

ℓ)1003816100381610038161003816 120597

minus119879ℓ+1⟩ 120575

ℓℓ1015840

⟨1198783|119860 (119890 119878)| 120597119860cork⟩ = sum

ℓ

⟨120597minus119879ℓ

1003816100381610038161003816119860ℓ(119890 119878

ℓ)1003816100381610038161003816 120597

minus119879ℓ+1⟩

(32)


⟨120597minus1198790

10038161003816100381610038161198600(119890 119878

0)1003816100381610038161003816 120597

minus1198791⟩ = 119886

2

0 (33)






3-spheres Σ0 Σ


intΣ01

3

119877radicℎ1198893119909 = 119871 sdot CS (Σ

01) (34)


1denoted by


01) (defined



119871 sdot CS (Σ01 119860) = 119871

3sdotCS (Σ

01)

1198712= int

Σ01

CS (Σ01)

1198712radicℎ119889

3119909

(35)

by using

1198713= vol (Σ

01) = int

Σ01

radicℎ1198893119909 (36)

Together with

3119877 =

3119896

1198862 (37)


3119896

1198862=CS (Σ

01)

1198712 (38)


120599 =1198862

1198712=

3

CS (Σ01) (39)


1198891199042= 119889119905

2minus 119886(119905)

2ℎ119894119896119889119909

119894119889119909

119896 (40)


(119886

119886)

2

=1

1198712 (41)


1 Σ

2) between Σ

1and Σ

2we have two cases


1(say 119896(Σ

1) = +1)


2(say 119896(Σ



1 Σ




119905 997891997888rarr +1 0 le 119905



1 Σ

2) is compact)


1198891198862=1198862

1198712119889119905

2= 120599119889119905

2 (43)





119899+1at level 119899 + 1

1198891198862|119899+1

sim 120599 sdot 119889119886210038161003816100381610038161003816119899

(44)


2+ 119889119910

2)119910


2119910


119889119886210038161003816100381610038161003816119899+1

=120599

119899 + 1sdot 119889119886

210038161003816100381610038161003816119899=

120599119899+1

(119899 + 1)sdot 119889119886

2100381610038161003816100381610038160 (45)


1198862=

infin

sum

119899=0

(119889119886210038161003816100381610038161003816119899) = 119886

2

0sdot

infin

sum

119899=0

1

119899120599119899

= 1198862

0sdot exp (120599) = 1198862

0sdot 119897scale

(46)


with 1198891198862|0= 119886

2



⟨1198783|119860 (119890 119878)| 120597119860cork⟩

= ((prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3)

times

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩)

times(prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3))

minus1

)

=

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

(47)


⟨1198783|119860 (119890 119878)| 120597119860cork⟩ =

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

= 1198862

0sdot

infin

sum

119899=0

1

119899(

3

CS (120597119860cork))

119899

= 1198862

0sdot exp( 3

CS (120597119860cork))

(48)



1 From the




0and Σ



Σ1

Ψ

997888rarr R

120601 darr 119894119889

Σ0

120595

997888rarr R

(49)


0and Σ







1)


0to the



1)


minus10038171003817100381710038171206011003817100381710038171003817

2= minus (120601

2

1+ 120601

2

2+ 120601

2

3) 997888rarr minus119890

minus21206011 (1 + 1206012

2+ 120601

2

3) (50)


1 Σ


0rarr


1 Σ

1rarr R and


0 Σ

1) as


119871 = 119877 + (120597120583120601)

2

minus120588

2(1 minus exp (minus120582120601))2 (51)




119871 = 119877 +1

611987221198772 (52)





1198783 cork997888rarr 120597119860cork

tower997888rarr 119875119875 (53)



119886 = 1198860sdot exp( 3

2 sdot CS (120597119860cork)+

3

2 sdot CS (119875119875)) (54)


⟨1198783|119860 (119890 119878)| 119875119875⟩ = 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(55)


⟨1198783timesR |119860 (119890 119878)| 119878

3timesR⟩

= sum


3times120579R |119860 (119890 119878)| 119878

3times120579R⟩

= 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(56)




1198783997888rarr 120597119860cork (57)


1198860= 119871

119875= radic

ℎ119866

1198883 (58)


119886 = 119871119875sdot exp( 3

2 sdot CS(Σ (2 5 7)) (59)


119871119875sdot exp(140

3) asymp 75 sdot 10

minus15119898 (60)


1198783997888rarr Σ (2 5 7) 997888rarr 119875119875 (61)


119886 = 119871119875sdot exp(140

3+ 90) asymp 914 sdot 10

24119898 asymp 10

9119871119895 (62)





120599 =3

2 sdot CS (Σ (2 5 7)) (63)



(1 + 120599 + (12059922) + (12059936)) (64)



=1

(1 + 120599 + (12059922) + (12059936))asymp 55325 sdot 10

minus5

(65)







Λ =1

1198712

119875

sdot exp(minus 3


3

CS (119875119875)) (67)


Λ sdot 1198712



minus118 (68)



0(expressing the


1198712

119888=

1198882

31198672

0

(69)


ΩΛ=

1198885

3ℎ1198661198672

0

sdot exp(minus 3


3

CS (119875119875)) (70)


1198670= 74

km119904 sdotMpc

(71)


ΩΛ= 0729 (72)


(1198670)Planck = 68

km119904 sdotMpc

(73)

for which we obtain

ΩΛasymp 088 (74)


(ΩΛ)Planck = 0683 (75)


119878 =119860 sdot 120574

0

4 sdot 120574 sdot 1198712

119875

sdot 2120587 (76)

with

1205740=


(77)


119875


1

1198712

119875

997888rarr1

1198712

119875


=2120587120574

0

1198712

119875

asymp 080037 sdot1

1198712

119875

(78)


ΩΛ=

1198885

3ℎ1198661198672

0



3

CS (119875119875))

(79)


ΩΛasymp 0704 (80)


[52]

1205740=


(81)


ΩΛasymp 06836 (82)


7 Conclusion


Appendices








119865119860= 119889119860 + 119860 and 119860 (B1)


and Chern class

1198622=

1

81205872int119872

tr (119865119860and 119865

119860) (B2)


int119872

tr (119865119860and 119865

119860) = int

120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B3)


CS (120597119872119860) = 1

81205872int120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B4)





119896 =1

241205872int120597119872

(119892minus1119889119892)

3

isin Z (B6)


CS (120597119872119860) mod 1 (B7)


120575CS (120597119872119860) = 0 (B8)


119889119860 + 119860 and 119860 = 0 (B9)



120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (B10)


120591 (Σ) =1

81205872intΣtimesR

tr (119865119860and 119865

119860) (B11)


1

81205872intΣtimesR

tr (119865119860and 119865

119860)

10038161003816100381610038161003816100381610038161003816

le1

81205872intΣtimesR

tr (119865119860and lowast119865

119860) (B12)


119860 Thus the






119897

119902+119898

119903lt 1 119897 mod 2 = 119898 mod 2

119896

119901+119897

119902+119898

119903gt 1

119896

119901minus119897

119902+119898

119903lt 1

119896

119901+119897

119902minus119898

119903lt 1

(C1)


CS (120572) = 1198902


with




⟨1 1 1⟩CS = 1

120

⟨1 1 3⟩CS = 49

120

(C4)


⟨1 1 3⟩CS = 81

280

⟨1 3 1⟩CS = 9

280

⟨1 2 2⟩CS = 169280

⟨1 2 4⟩CS = 249280

(C5)


120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (C6)


120591 (119875) =1

120 120591 (Σ (2 5 7)) =

9

280 (C7)

And we are done




Acknowledgment


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






3-spheres Σ0 Σ


intΣ01

3

119877radicℎ1198893119909 = 119871 sdot CS (Σ

01) (34)


1denoted by


01) (defined



119871 sdot CS (Σ01 119860) = 119871

3sdotCS (Σ

01)

1198712= int

Σ01

CS (Σ01)

1198712radicℎ119889

3119909

(35)

by using

1198713= vol (Σ

01) = int

Σ01

radicℎ1198893119909 (36)

Together with

3119877 =

3119896

1198862 (37)


3119896

1198862=CS (Σ

01)

1198712 (38)


120599 =1198862

1198712=

3

CS (Σ01) (39)


1198891199042= 119889119905

2minus 119886(119905)

2ℎ119894119896119889119909

119894119889119909

119896 (40)


(119886

119886)

2

=1

1198712 (41)


1 Σ

2) between Σ

1and Σ

2we have two cases


1(say 119896(Σ

1) = +1)


2(say 119896(Σ



1 Σ




119905 997891997888rarr +1 0 le 119905



1 Σ

2) is compact)


1198891198862=1198862

1198712119889119905

2= 120599119889119905

2 (43)





119899+1at level 119899 + 1

1198891198862|119899+1

sim 120599 sdot 119889119886210038161003816100381610038161003816119899

(44)


2+ 119889119910

2)119910


2119910


119889119886210038161003816100381610038161003816119899+1

=120599

119899 + 1sdot 119889119886

210038161003816100381610038161003816119899=

120599119899+1

(119899 + 1)sdot 119889119886

2100381610038161003816100381610038160 (45)


1198862=

infin

sum

119899=0

(119889119886210038161003816100381610038161003816119899) = 119886

2

0sdot

infin

sum

119899=0

1

119899120599119899

= 1198862

0sdot exp (120599) = 1198862

0sdot 119897scale

(46)


with 1198891198862|0= 119886

2



⟨1198783|119860 (119890 119878)| 120597119860cork⟩

= ((prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3)

times

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩)

times(prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3))

minus1

)

=

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

(47)


⟨1198783|119860 (119890 119878)| 120597119860cork⟩ =

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

= 1198862

0sdot

infin

sum

119899=0

1

119899(

3

CS (120597119860cork))

119899

= 1198862

0sdot exp( 3

CS (120597119860cork))

(48)



1 From the




0and Σ



Σ1

Ψ

997888rarr R

120601 darr 119894119889

Σ0

120595

997888rarr R

(49)


0and Σ







1)


0to the



1)


minus10038171003817100381710038171206011003817100381710038171003817

2= minus (120601

2

1+ 120601

2

2+ 120601

2

3) 997888rarr minus119890

minus21206011 (1 + 1206012

2+ 120601

2

3) (50)


1 Σ


0rarr


1 Σ

1rarr R and


0 Σ

1) as


119871 = 119877 + (120597120583120601)

2

minus120588

2(1 minus exp (minus120582120601))2 (51)




119871 = 119877 +1

611987221198772 (52)





1198783 cork997888rarr 120597119860cork

tower997888rarr 119875119875 (53)



119886 = 1198860sdot exp( 3

2 sdot CS (120597119860cork)+

3

2 sdot CS (119875119875)) (54)


⟨1198783|119860 (119890 119878)| 119875119875⟩ = 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(55)


⟨1198783timesR |119860 (119890 119878)| 119878

3timesR⟩

= sum


3times120579R |119860 (119890 119878)| 119878

3times120579R⟩

= 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(56)




1198783997888rarr 120597119860cork (57)


1198860= 119871

119875= radic

ℎ119866

1198883 (58)


119886 = 119871119875sdot exp( 3

2 sdot CS(Σ (2 5 7)) (59)


119871119875sdot exp(140

3) asymp 75 sdot 10

minus15119898 (60)


1198783997888rarr Σ (2 5 7) 997888rarr 119875119875 (61)


119886 = 119871119875sdot exp(140

3+ 90) asymp 914 sdot 10

24119898 asymp 10

9119871119895 (62)





120599 =3

2 sdot CS (Σ (2 5 7)) (63)



(1 + 120599 + (12059922) + (12059936)) (64)



=1

(1 + 120599 + (12059922) + (12059936))asymp 55325 sdot 10

minus5

(65)







Λ =1

1198712

119875

sdot exp(minus 3


3

CS (119875119875)) (67)


Λ sdot 1198712



minus118 (68)



0(expressing the


1198712

119888=

1198882

31198672

0

(69)


ΩΛ=

1198885

3ℎ1198661198672

0

sdot exp(minus 3


3

CS (119875119875)) (70)


1198670= 74

km119904 sdotMpc

(71)


ΩΛ= 0729 (72)


(1198670)Planck = 68

km119904 sdotMpc

(73)

for which we obtain

ΩΛasymp 088 (74)


(ΩΛ)Planck = 0683 (75)


119878 =119860 sdot 120574

0

4 sdot 120574 sdot 1198712

119875

sdot 2120587 (76)

with

1205740=


(77)


119875


1

1198712

119875

997888rarr1

1198712

119875


=2120587120574

0

1198712

119875

asymp 080037 sdot1

1198712

119875

(78)


ΩΛ=

1198885

3ℎ1198661198672

0



3

CS (119875119875))

(79)


ΩΛasymp 0704 (80)


[52]

1205740=


(81)


ΩΛasymp 06836 (82)


7 Conclusion


Appendices








119865119860= 119889119860 + 119860 and 119860 (B1)


and Chern class

1198622=

1

81205872int119872

tr (119865119860and 119865

119860) (B2)


int119872

tr (119865119860and 119865

119860) = int

120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B3)


CS (120597119872119860) = 1

81205872int120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B4)





119896 =1

241205872int120597119872

(119892minus1119889119892)

3

isin Z (B6)


CS (120597119872119860) mod 1 (B7)


120575CS (120597119872119860) = 0 (B8)


119889119860 + 119860 and 119860 = 0 (B9)



120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (B10)


120591 (Σ) =1

81205872intΣtimesR

tr (119865119860and 119865

119860) (B11)


1

81205872intΣtimesR

tr (119865119860and 119865

119860)

10038161003816100381610038161003816100381610038161003816

le1

81205872intΣtimesR

tr (119865119860and lowast119865

119860) (B12)


119860 Thus the






119897

119902+119898

119903lt 1 119897 mod 2 = 119898 mod 2

119896

119901+119897

119902+119898

119903gt 1

119896

119901minus119897

119902+119898

119903lt 1

119896

119901+119897

119902minus119898

119903lt 1

(C1)


CS (120572) = 1198902


with




⟨1 1 1⟩CS = 1

120

⟨1 1 3⟩CS = 49

120

(C4)


⟨1 1 3⟩CS = 81

280

⟨1 3 1⟩CS = 9

280

⟨1 2 2⟩CS = 169280

⟨1 2 4⟩CS = 249280

(C5)


120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (C6)


120591 (119875) =1

120 120591 (Σ (2 5 7)) =

9

280 (C7)

And we are done




Acknowledgment


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




with 1198891198862|0= 119886

2



⟨1198783|119860 (119890 119878)| 120597119860cork⟩

= ((prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3)

times

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩)

times(prod

ℓ

119885 (120597minus119879ℓ) 119885 (120597119860cork) 119885 (119878

3))

minus1

)

=

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

(47)


⟨1198783|119860 (119890 119878)| 120597119860cork⟩ =

infin

sum

119899=0

⟨120597minus119879119899

1003816100381610038161003816119860119899(119890 119878

119899)1003816100381610038161003816 120597

minus119879119899+1⟩

= 1198862

0sdot

infin

sum

119899=0

1

119899(

3

CS (120597119860cork))

119899

= 1198862

0sdot exp( 3

CS (120597119860cork))

(48)



1 From the




0and Σ



Σ1

Ψ

997888rarr R

120601 darr 119894119889

Σ0

120595

997888rarr R

(49)


0and Σ







1)


0to the



1)


minus10038171003817100381710038171206011003817100381710038171003817

2= minus (120601

2

1+ 120601

2

2+ 120601

2

3) 997888rarr minus119890

minus21206011 (1 + 1206012

2+ 120601

2

3) (50)


1 Σ


0rarr


1 Σ

1rarr R and


0 Σ

1) as


119871 = 119877 + (120597120583120601)

2

minus120588

2(1 minus exp (minus120582120601))2 (51)




119871 = 119877 +1

611987221198772 (52)





1198783 cork997888rarr 120597119860cork

tower997888rarr 119875119875 (53)



119886 = 1198860sdot exp( 3

2 sdot CS (120597119860cork)+

3

2 sdot CS (119875119875)) (54)


⟨1198783|119860 (119890 119878)| 119875119875⟩ = 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(55)


⟨1198783timesR |119860 (119890 119878)| 119878

3timesR⟩

= sum


3times120579R |119860 (119890 119878)| 119878

3times120579R⟩

= 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(56)




1198783997888rarr 120597119860cork (57)


1198860= 119871

119875= radic

ℎ119866

1198883 (58)


119886 = 119871119875sdot exp( 3

2 sdot CS(Σ (2 5 7)) (59)


119871119875sdot exp(140

3) asymp 75 sdot 10

minus15119898 (60)


1198783997888rarr Σ (2 5 7) 997888rarr 119875119875 (61)


119886 = 119871119875sdot exp(140

3+ 90) asymp 914 sdot 10

24119898 asymp 10

9119871119895 (62)





120599 =3

2 sdot CS (Σ (2 5 7)) (63)



(1 + 120599 + (12059922) + (12059936)) (64)



=1

(1 + 120599 + (12059922) + (12059936))asymp 55325 sdot 10

minus5

(65)







Λ =1

1198712

119875

sdot exp(minus 3


3

CS (119875119875)) (67)


Λ sdot 1198712



minus118 (68)



0(expressing the


1198712

119888=

1198882

31198672

0

(69)


ΩΛ=

1198885

3ℎ1198661198672

0

sdot exp(minus 3


3

CS (119875119875)) (70)


1198670= 74

km119904 sdotMpc

(71)


ΩΛ= 0729 (72)


(1198670)Planck = 68

km119904 sdotMpc

(73)

for which we obtain

ΩΛasymp 088 (74)


(ΩΛ)Planck = 0683 (75)


119878 =119860 sdot 120574

0

4 sdot 120574 sdot 1198712

119875

sdot 2120587 (76)

with

1205740=


(77)


119875


1

1198712

119875

997888rarr1

1198712

119875


=2120587120574

0

1198712

119875

asymp 080037 sdot1

1198712

119875

(78)


ΩΛ=

1198885

3ℎ1198661198672

0



3

CS (119875119875))

(79)


ΩΛasymp 0704 (80)


[52]

1205740=


(81)


ΩΛasymp 06836 (82)


7 Conclusion


Appendices








119865119860= 119889119860 + 119860 and 119860 (B1)


and Chern class

1198622=

1

81205872int119872

tr (119865119860and 119865

119860) (B2)


int119872

tr (119865119860and 119865

119860) = int

120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B3)


CS (120597119872119860) = 1

81205872int120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B4)





119896 =1

241205872int120597119872

(119892minus1119889119892)

3

isin Z (B6)


CS (120597119872119860) mod 1 (B7)


120575CS (120597119872119860) = 0 (B8)


119889119860 + 119860 and 119860 = 0 (B9)



120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (B10)


120591 (Σ) =1

81205872intΣtimesR

tr (119865119860and 119865

119860) (B11)


1

81205872intΣtimesR

tr (119865119860and 119865

119860)

10038161003816100381610038161003816100381610038161003816

le1

81205872intΣtimesR

tr (119865119860and lowast119865

119860) (B12)


119860 Thus the






119897

119902+119898

119903lt 1 119897 mod 2 = 119898 mod 2

119896

119901+119897

119902+119898

119903gt 1

119896

119901minus119897

119902+119898

119903lt 1

119896

119901+119897

119902minus119898

119903lt 1

(C1)


CS (120572) = 1198902


with




⟨1 1 1⟩CS = 1

120

⟨1 1 3⟩CS = 49

120

(C4)


⟨1 1 3⟩CS = 81

280

⟨1 3 1⟩CS = 9

280

⟨1 2 2⟩CS = 169280

⟨1 2 4⟩CS = 249280

(C5)


120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (C6)


120591 (119875) =1

120 120591 (Σ (2 5 7)) =

9

280 (C7)

And we are done




Acknowledgment


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119886 = 1198860sdot exp( 3

2 sdot CS (120597119860cork)+

3

2 sdot CS (119875119875)) (54)


⟨1198783|119860 (119890 119878)| 119875119875⟩ = 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(55)


⟨1198783timesR |119860 (119890 119878)| 119878

3timesR⟩

= sum


3times120579R |119860 (119890 119878)| 119878

3times120579R⟩

= 1198862

0sdot exp( 3

CS (120597119860cork)+

3

CS (119875119875))

(56)




1198783997888rarr 120597119860cork (57)


1198860= 119871

119875= radic

ℎ119866

1198883 (58)


119886 = 119871119875sdot exp( 3

2 sdot CS(Σ (2 5 7)) (59)


119871119875sdot exp(140

3) asymp 75 sdot 10

minus15119898 (60)


1198783997888rarr Σ (2 5 7) 997888rarr 119875119875 (61)


119886 = 119871119875sdot exp(140

3+ 90) asymp 914 sdot 10

24119898 asymp 10

9119871119895 (62)





120599 =3

2 sdot CS (Σ (2 5 7)) (63)



(1 + 120599 + (12059922) + (12059936)) (64)



=1

(1 + 120599 + (12059922) + (12059936))asymp 55325 sdot 10

minus5

(65)







Λ =1

1198712

119875

sdot exp(minus 3


3

CS (119875119875)) (67)


Λ sdot 1198712



minus118 (68)



0(expressing the


1198712

119888=

1198882

31198672

0

(69)


ΩΛ=

1198885

3ℎ1198661198672

0

sdot exp(minus 3


3

CS (119875119875)) (70)


1198670= 74

km119904 sdotMpc

(71)


ΩΛ= 0729 (72)


(1198670)Planck = 68

km119904 sdotMpc

(73)

for which we obtain

ΩΛasymp 088 (74)


(ΩΛ)Planck = 0683 (75)


119878 =119860 sdot 120574

0

4 sdot 120574 sdot 1198712

119875

sdot 2120587 (76)

with

1205740=


(77)


119875


1

1198712

119875

997888rarr1

1198712

119875


=2120587120574

0

1198712

119875

asymp 080037 sdot1

1198712

119875

(78)


ΩΛ=

1198885

3ℎ1198661198672

0



3

CS (119875119875))

(79)


ΩΛasymp 0704 (80)


[52]

1205740=


(81)


ΩΛasymp 06836 (82)


7 Conclusion


Appendices








119865119860= 119889119860 + 119860 and 119860 (B1)


and Chern class

1198622=

1

81205872int119872

tr (119865119860and 119865

119860) (B2)


int119872

tr (119865119860and 119865

119860) = int

120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B3)


CS (120597119872119860) = 1

81205872int120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B4)





119896 =1

241205872int120597119872

(119892minus1119889119892)

3

isin Z (B6)


CS (120597119872119860) mod 1 (B7)


120575CS (120597119872119860) = 0 (B8)


119889119860 + 119860 and 119860 = 0 (B9)



120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (B10)


120591 (Σ) =1

81205872intΣtimesR

tr (119865119860and 119865

119860) (B11)


1

81205872intΣtimesR

tr (119865119860and 119865

119860)

10038161003816100381610038161003816100381610038161003816

le1

81205872intΣtimesR

tr (119865119860and lowast119865

119860) (B12)


119860 Thus the






119897

119902+119898

119903lt 1 119897 mod 2 = 119898 mod 2

119896

119901+119897

119902+119898

119903gt 1

119896

119901minus119897

119902+119898

119903lt 1

119896

119901+119897

119902minus119898

119903lt 1

(C1)


CS (120572) = 1198902


with




⟨1 1 1⟩CS = 1

120

⟨1 1 3⟩CS = 49

120

(C4)


⟨1 1 3⟩CS = 81

280

⟨1 3 1⟩CS = 9

280

⟨1 2 2⟩CS = 169280

⟨1 2 4⟩CS = 249280

(C5)


120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (C6)


120591 (119875) =1

120 120591 (Σ (2 5 7)) =

9

280 (C7)

And we are done




Acknowledgment


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





0(expressing the


1198712

119888=

1198882

31198672

0

(69)


ΩΛ=

1198885

3ℎ1198661198672

0

sdot exp(minus 3


3

CS (119875119875)) (70)


1198670= 74

km119904 sdotMpc

(71)


ΩΛ= 0729 (72)


(1198670)Planck = 68

km119904 sdotMpc

(73)

for which we obtain

ΩΛasymp 088 (74)


(ΩΛ)Planck = 0683 (75)


119878 =119860 sdot 120574

0

4 sdot 120574 sdot 1198712

119875

sdot 2120587 (76)

with

1205740=


(77)


119875


1

1198712

119875

997888rarr1

1198712

119875


=2120587120574

0

1198712

119875

asymp 080037 sdot1

1198712

119875

(78)


ΩΛ=

1198885

3ℎ1198661198672

0



3

CS (119875119875))

(79)


ΩΛasymp 0704 (80)


[52]

1205740=


(81)


ΩΛasymp 06836 (82)


7 Conclusion


Appendices








119865119860= 119889119860 + 119860 and 119860 (B1)


and Chern class

1198622=

1

81205872int119872

tr (119865119860and 119865

119860) (B2)


int119872

tr (119865119860and 119865

119860) = int

120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B3)


CS (120597119872119860) = 1

81205872int120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B4)





119896 =1

241205872int120597119872

(119892minus1119889119892)

3

isin Z (B6)


CS (120597119872119860) mod 1 (B7)


120575CS (120597119872119860) = 0 (B8)


119889119860 + 119860 and 119860 = 0 (B9)



120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (B10)


120591 (Σ) =1

81205872intΣtimesR

tr (119865119860and 119865

119860) (B11)


1

81205872intΣtimesR

tr (119865119860and 119865

119860)

10038161003816100381610038161003816100381610038161003816

le1

81205872intΣtimesR

tr (119865119860and lowast119865

119860) (B12)


119860 Thus the






119897

119902+119898

119903lt 1 119897 mod 2 = 119898 mod 2

119896

119901+119897

119902+119898

119903gt 1

119896

119901minus119897

119902+119898

119903lt 1

119896

119901+119897

119902minus119898

119903lt 1

(C1)


CS (120572) = 1198902


with




⟨1 1 1⟩CS = 1

120

⟨1 1 3⟩CS = 49

120

(C4)


⟨1 1 3⟩CS = 81

280

⟨1 3 1⟩CS = 9

280

⟨1 2 2⟩CS = 169280

⟨1 2 4⟩CS = 249280

(C5)


120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (C6)


120591 (119875) =1

120 120591 (Σ (2 5 7)) =

9

280 (C7)

And we are done




Acknowledgment


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




and Chern class

1198622=

1

81205872int119872

tr (119865119860and 119865

119860) (B2)


int119872

tr (119865119860and 119865

119860) = int

120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B3)


CS (120597119872119860) = 1

81205872int120597119872

tr(119860 and 119889119860 + 23119860 and 119860 and 119860) (B4)





119896 =1

241205872int120597119872

(119892minus1119889119892)

3

isin Z (B6)


CS (120597119872119860) mod 1 (B7)


120575CS (120597119872119860) = 0 (B8)


119889119860 + 119860 and 119860 = 0 (B9)



120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (B10)


120591 (Σ) =1

81205872intΣtimesR

tr (119865119860and 119865

119860) (B11)


1

81205872intΣtimesR

tr (119865119860and 119865

119860)

10038161003816100381610038161003816100381610038161003816

le1

81205872intΣtimesR

tr (119865119860and lowast119865

119860) (B12)


119860 Thus the






119897

119902+119898

119903lt 1 119897 mod 2 = 119898 mod 2

119896

119901+119897

119902+119898

119903gt 1

119896

119901minus119897

119902+119898

119903lt 1

119896

119901+119897

119902minus119898

119903lt 1

(C1)


CS (120572) = 1198902


with




⟨1 1 1⟩CS = 1

120

⟨1 1 3⟩CS = 49

120

(C4)


⟨1 1 3⟩CS = 81

280

⟨1 3 1⟩CS = 9

280

⟨1 2 2⟩CS = 169280

⟨1 2 4⟩CS = 249280

(C5)


120591 (Σ) = min CS (120572) | 120572 1205871(Σ) 997888rarr SU (2) (C6)


120591 (119875) =1

120 120591 (Σ (2 5 7)) =

9

280 (C7)

And we are done




Acknowledgment


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






Acknowledgment


References






























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics
























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Research Article Inflation and Topological Phase...

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