torsten asselmeyer-maluga and jerzy krol- abelian gerbes, generalized geometries and exotic r^4
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Abelian gerbes, generalized geometries and exotic R4
Torsten Asselmeyer-Maluga1 and Jerzy Krol21)German Aero space center, Rutherfordstr. 2, 12489 Berlin, Germanya)
2)University of Silesia, ul. Uniwesytecka 4, 40-007 Katowice, Polandb)
(Dated: September 1, 2011)
In the paper we prove the existence of the strict relation between small exotic smoothness structures onthe Euclidean 4-space R4 from the radial family of DeMichelis-Freedman type, and cobordism classes ofcodimension one foliations of S3. Both are distinguished by the Godbillon-Vey invariants, GV ∈ H3(S3,R),of the foliations which are computed from the value of radii of the radial family. The special case of integerGodbillon-Vey invariants GV ∈ H3(S3,Z) is also discussed and is connected to flat PSL(2,R)−bundles.Next we relate these distinguished small exotic smooth R4’s with twisted generalized geometries of Hitchinon TS3 ⊕ T ⋆S3 and abelian gerbes on S3. In particular the change of the smoothness on R4 corresponds tothe twisting of the generalized geometry by the abelian gerbe. We formulate the localization principle forexotic 4-regions in spacetime and show that the existence of such domains causes the quantization of electriccharge, the effect usually ascribed to the existence of magnetic monopoles.
PACS numbers: 02.40.Sf, 02.40.Vh, 04.20.GzKeywords: exotic smoothness, foliations, gerbe, Hitchin structure, monopole
CONTENTS
I. Introduction 1
II. Preliminaries: Exotic Smoothness ofmanifolds 3A. Smoothness on manifolds 3B. Exotic smooth 4-manifolds and Akbulut
corks 3
III. Preliminaries: Codimension-one foliationson 3-manifolds 4A. Definition of Foliation and foliated
cobordism 4B. Non-cobordant foliations of S3 detected by
the Godbillon-Vey class 6C. Codimension-one foliations on 3-manifolds 7
IV. Exotic R4 and codimension-onefoliations 8A. Exotic R4 and Casson handles 8B. The design of a Casson handle and its
foliation 9C. Capped gropes and its design 10D. The radial family of uncountably infinite
small exotic R4 11E. Exotic R4 and codimension-1 foliations 11F. Integer Godbillon-Vey invariants and flat
bundles 13G. The conjecture: the failure of the smooth
4-dimensional Poincare conjecture 14
V. Abelian gerbes on S3 14
a)Electronic mail: [email protected])Electronic mail: [email protected]
VI. Generalized (complex) structures ofHitchin 15
VII. An application: charge quantizationwithout magnetic monopoles 18A. Dirac’s magnetic monopoles and S1 -
gerbes 18B. S1-gerbes on S3 and exotic smooth R4’s 19
Acknowledgment 20
References 20
I. INTRODUCTION
If one formally considers the path integral over space-time geometries then one has to include the possibility ofdifferent smooth structures for spacetime8,72. Brans22–24
was the first who considered exotic smoothness also onopen 4-manifolds as a possibility for space-time. He con-jectured that exotic smoothness induces an additionalgravitational field (Brans conjecture). The conjecturewas established by Asselmeyer7 in the compact case andby S ladkowski76 in the non-compact case. But there is abig problem which prevents progress in the understand-ing of exotic smoothness especially for the R4: there is noknown explicit coordinate representation. As the resultno exotic smooth function on any such R4 is known eventhough there exist families of infinite continuum manydifferent non diffeomorphic smooth R4. This is also astrong limitation for the applicability to physics of non-standard open 4-smoothness. Bizaca18 was able to con-struct an infinite cover of local coordinate patches by us-ing Casson handles. But it still seems hopeless to extractphysical information from that approach.
In this paper we address this issue. We choose another,relative way of tracing the changes of smoothness struc-
2
tures especially on the simplest Euclidean 4-manifold, i.e.R4. For that purpose we have to consider the technicaltool of h-cobordism for 4-manifolds. Fortunately there isa structure theorem for such h-cobordisms reflecting thedifference in the smoothness for two non-diffeomorphicbut homeomorphic 4-manifolds: both manifolds differ bya contractable submanifold (with boundary) called anAkbulut cork61. A careful analysis of that example leadsus to the amazing relation between smoothness structureson 4-manifold and codimension-1 foliations of the bound-ary for the Akbulut cork. In case of the exotic R4, one canuse the work of Bizaca to construct something like an Ak-bulut cork64 inside of the R4. To be precise, one considersan Akbulut cork and attach a Casson handle to it. Theinterior of this construction is mostly an exotic small R4.Considering this example, DeMichelis and Freedman29
constructed continuous family of nondiffeomorphic smallexotic R4, the so-called radial family. This parametriza-tion will be used to construct a relation to noncobordant,codimension-1 Thurston-type foliations of the boundaryΣ = ∂A of the Akbulut cork A (see Theorem 5). Thisfoliation is partly classified by the Godbillon-Vey number(with respect to foliated cobordism). From the surgeryconstruction of Σ, we obtain a Thurston-type foliation onS3 with the same Godbillon-Vey number (see Theorem4). The smoothness structure of a 3-manifold is unique(up to diffeomorphism), i.e. an exotic smoothness struc-ture on a 4-manifold with boundary cannot be detectedby the smoothness structure of its boundary. But withthe theorems 5 and 8 we found some other structures on3-manifolds which are powerful enough to do it. To bemore precise, the changes of various structures on S3,like S1-gerbes or generalized geometries of Hitchin, areable to reflect the changes of smoothness on 4-manifoldslikeR4, assuming S3 lies at the boundary of the Akbulutcork (or better is contained inside of the cork) used toconstruct this exotic R4. Even though smooth manifoldsare 4-dimensional, the structures on S3 refer, in general,to 2d conformal field theory (CFT), abelian gerbes on S3,C⋆-algebras, quantization12, or with some D- and NS-branes configurations in 10d string theory11,13–15 . Thisobservation indicates that exotic R4’s may play a distin-guished geometrical role for quantum gravity which ex-tends the role of standard smooth R4 in the Riemanniangeometry and classical gravity (see also8,16,30,52).
Having this in mind, we deal with the details of thefundamental relation of small exotic R4 from the radialfamily of DeMichelis-Freedman, and codimension one fo-liations of certain 3-sphere seen as the part of the bound-ary of the Akbulut cork. This is new and importantmathematical result. It enables one to describe exoticR4 in terms of obstructions to become standard smooth.Various structures on S3 emerge which can be consideredas the obstructions to standardness. However, the exis-tence of such 3-sphere has far reaching consequences alsofor physics. In different limits of a given theory differ-ent obstruction data become important. We formulatethe localization principle which relies on the localization
of effects of small exotic R4’s from the radial family in4-spacetime, giving the geometric realization of the cor-responding classes in H3(S3,Z) in this spacetime. Inparticular the effects of this localization are the same asthose of magnetic monopoles. Then the quantization ofthe electric charge without magnetic monopoles follows.
Further work has to be done to uncover fully the entirephysical meaning of the localization principle for exoticopen smooth manifolds. This paper serves as a crucialtechnical step only. Especially there is the interestingquestion to find an analogous characterization of largeexotic R4’s, i.e. those which contain a compact set non-embeddable in standard R4. The techniques presentedhere seem to be applicable for this case too, though wehave to take some further deformations of our test space,i.e. S3, to detect correctly the changes in large smooth-ness on R4. The possible indications come from thecategorical approach to gerbes and related generalizedquotients of spaces, as well from geometries of Gualtieri-Hitchin again.
The paper is organized as follows. In the next twosections we present a short introduction into smooth-ness structures on manifolds and codimension-1 folia-tions. Then we will derive the relation between exoticR4 and codimension-1 foliations of S3 and obtain themain result (Theorem 5) of the paper:The exotic R4 from the radial family of DeMichelis-Freedman is determined by the codimension-1 folia-tion with non-vanishing Godbillon-Vey (GV) class inH3(S3,R3) of a 3-sphere seen as submanifold S3 ⊂ R4.Conversely, the codimension-1 foliation on S3 with non-vanishing GV number r2 ∈ R, is determined by an exoticsmooth R4 in the radial family which corresponds to thevalue r of the radii.Moreover, we give direct characterization of thecodimension-1 foliations of S3 with integer GV number(Theorem 6): These correspond to the flat PSL(2,R)bundles on M = S2 \ k − punctures × S1 and the GVinvariant of this foliation (when evaluated on the funda-mental class of M) is equal ±(2 − k). In the subsectionIV G we state the conjecture leading to exotic smoothnesson S4: Every 3-sphere admitting codimension-1 foliationwith integer GV class bounds an exotic 4-disk. The glu-ing of this exotic 4-disk to the standard 4-disk gives anexotic S4. Next in Sec. V we present a short introductioninto abelian gerbes and the relation of the exotic R4 (de-rived from the codimension-1 foliations of S3 with integerGV classes) to abelian gerbes on the 3-sphere S3. Thisallows us in Sec. VI for assigning the changes of smooth-ness on R4 to twistings of the generalized geometries onTS3 ⊕ T ⋆S3 by abelian gerbes. Based on these resultswe will show in the last section that the appearance ofsmall exotic R4 in 4-spacetime can be used to explainthe quantization of electric charge but without magneticmonopoles. Mathematical results obtained in this paperare of great potential value for the applications to physicsand especially in string theory. It will be further workedout in the forthcoming papers (but see the first results
3
in7–10,12,17,52,53,76).
II. PRELIMINARIES: EXOTIC SMOOTHNESS OFMANIFOLDS
A. Smoothness on manifolds
Einstein’s insight that gravity is the manifestation ofgeometry leads to a new view on the structure of space-time. From the mathematical point of view, spacetime isa smooth 4-manifold endowed with a (smooth) metric asbasic variable for general relativity. Later on, the exis-tence question for Lorentz structure and causality prob-lems (see Hawking and Ellis44) gave further restrictionson the 4-manifold: causality implies non-compactness,Lorentz structure needs a codimension-1 foliation. Usu-ally, one starts with a globally foliated, non-compact 4-manifold Σ × R fulfilling all restrictions where Σ is asmooth 3-manifold representing the spatial part. Butother non-compact 4-manifolds are also possible, i.e. itis enough to assume that a non-compact, smooth 4-manifold be endowed with a codimension-1 foliation.
All these restrictions on the representation of space-time by the manifold concept are clearly motivated byphysical questions. Among the properties there is onedistinguished element: the smoothness. Usually one as-sumes a smooth, unique atlas of charts (i.e. a smoothor differential structure) covering the manifold where thesmoothness is induced by the unique smooth structureon R. But that is not the full story. Even in dimen-sion 4, there are an infinity of possible other smoothnessstructures (i.e. a smooth atlas) non-diffeomorphic to eachother. For a deeper insight we refer to the book9.
If two manifolds are homeomorphic but non-diffeomorphic, they are exotic to each other. Thesmoothness structure is called an exotic smoothnessstructure.
The implications for physics are tremendous becausewe rely on the smooth calculus to formulate field theo-ries. Thus different smoothness structures have to repre-sent different physical situations leading to different mea-surable results. But it should be stressed that exoticsmoothness is not exotic physics. Exotic smoothness is amathematical structure which should be further exploredto understand its physical relevance.
Usually one starts with a topological manifold M andintroduces structures on them. Then one has the follow-ing ladder of possible structures:
Topology → piecewise-linear(PL) → Smoothness
→bundles, Lorentz, Spin etc. → metric,...
We do not want to discuss the first transition, i.e. theexistence of a triangulation on a topological manifold.But we remark that the existence of a PL structure im-plies uniquely a smoothness structure in all dimensionssmaller than 750.
Given two homeomorphic manifolds M,M ′, how canwe compare both manifolds to decide whether both arediffeomorphic? A mapping between two manifolds M,M ′
can be described by a (n + 1)−dimensional manifold Wwith ∂W = M ⊔ M ′, called a cobordism W . In thefollowing, the two homeomorphic manifolds M,M ′ aresimple-connected. The celebrated h-cobordism theoremof Smale77,78 gives a simple criteria when a diffeomor-phism between M and M ′ for dimension greater than5 exists: there is a cobordism W between M and M ′
where the inclusions M,M ′ → W induce homotopy-equivalences between M,M ′ and W . We call W ah-cobordism. Therefore the classification problem ofsmoothness structures in higher dimensions (> 5) is ahomotopy-theoretic problem. The set of smooth struc-tures (up to h-cobordisms) of the n−sphere Sn has thestructure of an abelian group Θn (under connected sum#). Via the h-cobordism theorem this group is identicalto the group of homotopy spheres which was analyzed byKervaire and Milnor48 to be a finite group. Later on theresult about the exotic spheres can be extended to anysmoothable manifold50, i.e. there is only a finite num-ber of non-diffeomorphic smoothness structures. In alldimensions smaller than four, there is an unique smooth-ness structure, the standard structure. But all methodsfailed for the special case of dimension four.
B. Exotic smooth 4-manifolds and Akbulut corks
Now we consider two homeomorphic, smooth, but non-diffeomorphic 4-manifolds M0 and M . As expressedabove, a comparison of both smoothness structures isgiven by a h-cobordism W between M0 and M (M,M0
are homeomorphic). Let the 4-manifolds additionally becompact, closed and simple-connected, then we have thestructure theorem65 of h-cobordisms28:
Theorem 1 Let W be a h-cobordisms between M0, M ,then there are contractable submanifolds A0 ⊂ M0, A ⊂M and a h subcobordism X ⊂ W with ∂X = A0 ⊔ A,so that the remaining h-cobordism W \X trivializes W \X = (M0\A0)× [0, 1] inducing a diffeomorphism betweenM0 \A0 and M \A.
In short that means that the smoothness structure of Mis determined by the contractable manifold A – its Akbu-lut cork – and by the embedding of A into M . As shownby Freedman34, the Akbulut cork has a homology 3-sphere66 as boundary. The embedding of the cork can bederived now from the structure of the h-subcobordism Xbetween A0 and A. For that purpose we cut A0 out fromM0 and A out from M . Then we glue in both submani-folds A0, A via the maps τ0 : ∂A0 → ∂(M0 \ A0) = ∂A0
and τ : ∂A → ∂(M \ A) = ∂A. Both maps τ0, τ are in-volutions, i.e. τ τ = id. One of these maps (say τ0) canbe chosen to be trivial (say τ0 = id). Thus the involu-tion τ determines the smoothness structure. Especiallythe topology of the Akbulut cork A and its boundary ∂A
4
is given by the topology of M . For instance, the Ak-bulut cork of the blow-uped 4-dimensional K3 surface
K3#CP2
is the so-called Mazur manifold1,2 with theBrieskorn-Sphere B(2, 5, 7) as boundary. Akbulut andits coworkers3,4 discuss many examples of Akbulut corksand the dependence of the smoothness structure on thecork.
For the following we need a short account of the proofof the h-cobordism structure theorem. The interior ofevery h-cobordism can be divided into pieces, calledhandle58. A k-handle is the manifold Dk ×D5−k whichwill be glued along the boundary Sk ×D5−k. The pairsof 0−/1− and 4−/5−handles in a h-cobordism betweenthe two homeomorphic 4-manifolds M0 and M can bekilled by a general procedure58. Thus only the pairs of2 − /3−handles are left. Exactly these pairs are the dif-ference between the smooth h-cobordism and the topo-logical h-cobordism. To eliminate the 2 − /3−handlesone has to embed a disk without self-intersections intoM (Whitney trick). But that is mostly impossible in 4-dimensional manifolds. Therefore Casson27 constructedby an infinite, recursive process a special handle – theCasson-handle CH – containing the required disk with-out self-intersections. Freedman was able to show topo-logically the existence of this disk and he constructs34 ahomeomorphism between every Casson handle CH andthe open 2-handle D2 × R2. But CH is in general non-diffeomorphic to D2 ×R2 as shown later by Gompf40,41.
Now we consider the smooth h-cobordism W togetherwith a neighborhood N of 2 − /3−handles. It is enoughto take a pair of handles with two self-intersections (ofopposite orientation) between the 2- and 3-Spheres atthe boundary of the handle. Thus one can constructan Akbulut cork A in M out of this data28. The pairof 2− /3−handles can be eliminated topologically by theembedding of a Casson handle. Then as shown by Bizacaand Gompf20 the neighborhood N of the handle pair aswell the neighborhood N(A) of the embedded Akbulutcork consists of the cork A and the Casson handle CH .Especially the open neighborhood N(A) of the Akbulutcork is an exotic R4. The situation was analyzed in42:
Theorem 2 Let W 5 be a non-trivial (smooth) h-cobordism between M4
0 and M4 (i.e. W is not diffeomor-phic toM×[0, 1]). Then there is an open sub-h-cobordismU5 that is homeomorphic to R4 × [0, 1] and contains acompact contractable sub-h-cobordism X (the cobordismbetween the Akbulut corks, see above), such that both Wand U are trivial cobordisms outside of X, i.e. one hasthe diffeomorphisms
W \X = ((W ∩M) \X) × [0, 1] and
U \X = ((U ∩M) \X) × [0, 1]
(the latter can be chosen to be the restriction of the for-mer). Furthermore the open sets U ∩M and U ∩M0 arehomeomorphic to R4 which are exotic R4 if W is non-trivial.
Then one gets an exotic R4 which smoothly embeds au-tomatically in the 4-sphere, called a small exotic R4.Furthermore we remark that the exoticness of the R4 isconnected with non-triviality of the smooth h-cobordismW 5, i.e. the failure of the smooth h-cobordism theoremimplies the existence of small exotic R4’s.
III. PRELIMINARIES: CODIMENSION-ONEFOLIATIONS ON 3-MANIFOLDS
In short, a foliation of a smooth manifold M is anintegrable subbundle N ⊂ TM of the tangent bun-dle TM . The existence of codimension-1-foliations de-pends strongly on the compactness or non-compactnessof the manifold. Every compact manifold admits acodimension-1-foliation if and only if the Euler charac-teristics vanish. In the following we will first concentrateon the 3-sphere S3 with vanishing Euler characteristicsadmitting codimension-1-foliations.
A. Definition of Foliation and foliated cobordism
A codimension k foliation67 of an n-manifold Mn (seethe nice overview article54) is a geometric structure whichis formally defined by an atlas φi : Ui →Mn, withUi ⊂ Rn−k ×Rk, such that the transition functions havethe form
φij(x, y) = (f(x, y), g(y)),[x ∈ Rn−k, y ∈ Rk
].
To be more precise one has
Definition 1 Let (M,A) be an n−dimensional smoothmanifold (of differential structure A) with bound-ary: we take φλ : Uλ → Vλ ⊂ Rn+, Rn+ =(x1, . . . , xn) ∈ Rn| xn ≥ 0 for a chart (Uλ, φλ) ∈ A.Let k with 0 ≤ k ≤ n be an integer. Let F = Lα|α ∈ Abe a family of arcwise connected subsets Lα of the mani-fold M . We say that F is a k−dimensional smooth foli-ation of M if it satisfies the following rules.
1. Lα ∩ Lβ = ∅ for α, β ∈ A with α 6= β.
2.⋃α∈A(Lα) = M
3. Given a point p in M , there exists a chart(Uλ, φλ) ∈ A about p such that for Lα with Uλ ∩Lα 6= ∅, α ∈ A, each (arcwise) connected compo-nent of φλ(Uλ ∩ Lα) is of the form
(x1, . . . , xn) ∈ φλ(Uλ) | xk+1 = ck+1, . . . , xn = cn
where ck+1, ck+2, . . . , cn are constants determinedby the (arcwise) connected component.
4. For p ∈ ∂M ∩Lα we denote the boundary by ∂F =Lα |α ∈ A, Lα ⊂ ∂M
5
Figure 1. foliation of D1× S1
We call Lα a leaf of the foliation F . The foliation F isalso referred to as a smooth codimension n − k foliationor a smooth foliation of codimension n− k.
Intuitively, a foliation is a pattern of (n−k)-dimensionalstripes - i.e., submanifolds - on Mn, called the leaves ofthe foliation, which are locally well-behaved. The tan-gent space to the leaves of a foliation F forms a vectorbundle over Mn, denoted TF . The complementary bun-dle νF = TMn/TF is the normal bundle of F . Suchfoliations are called regular in contrast to singular folia-tions or Haefliger structures. For the important case of acodimension-1 foliation we need an overall non-vanishingvector field or its dual, an one-form ω. This one-formdefines a foliation iff it is integrable, i.e.
dω ∧ ω = 0 (1)
The cross-product M ×N defines a trivial foliation. Oneof the first examples of a nontrivial foliation is known asReeb foliation. Let f : (−1, 1) → R be a smooth functionf(t) = exp(t2/(1 − t2)) − 1. But every function f with
f(0) = 0 f(t) ≥ 0 f(t) = f(−t)limt→±1
dk
dtkf(t) = ∞ limt→±1
dk
dtk
(1
dfdt
(t)
)= 0
with −1 < t < 1 and k = 0, 1, 2, . . . will also work. Definesubsets L
′
α, 0 ≤ α < 1, and L′
± of D1 × S1 by
L′
α = t, exp (2π(α+ f(t))i) | − 1 < t < 1 ,L
′
± =
(±1, e2πθi)| 0 ≤ θ < 1
defining a smooth foliation of D1 × S1(see the Fig. 1).The join of two copies results in a smooth foliation of
the torus T 2 = (D1×S1)∪ (D1×S1). This example canbe generalized to the solid torus D2 ×S1 by defining thesubsets Lα by
Lα =x, exp (2π(α+ f(|x|))i) | x ∈ int(D2)
, 0 ≤ α < 1
where |x| is the distance between the origin of the discand the point x in the interior in(D2). The family of
sets Lα and L′
α, L′
± for the boundary ∂(D2 × S1) = T 2
Figure 2. Reeb foliation solid torus
forms a smooth foliation of D2 × S1 which is known asReeb foliation (see the Fig. 2). Pasting together twocopies of the solid torus (D2 ×S1)∪ (S1 ×D2) along thecommon boundary results in a 3-sphere. Then we obtainthe Reeb foliation of the 3-sphere. To complete the dis-cussion, we have to construct the 1-form ω together withthe integrability condition (1). Represent the points ofD2×S1 by the polar coordinates (r, ψ, y) ∈ D2×S1 with0 ≤ r ≤ 1, 0 ≤ ψ ≤ 2π, y ∈ S1. Consider a monotone de-creasing function h : [0, 1] → [0, 1] with limr→0 h(r) = 1
and limr→1 h(r) = 0 (for instance h(r) = exp(− r2
(1−r)2 )
will work). Define a 1-form ω on D2 × S1 by
ω =
−drh(r)
(dy − df
drdr)
+ (1 − h(r))((
1/ dfdr
)dy − dr
)
(2)on ∂D2×S1 or on int(D2)×S1, respectively, and we getthe relation
dω = −ω ∧ d(
log
(h(r) +
(1 + h(r))dfdr
))(3)
leading to the integrability condition ω ∧ dω = 0.Now we will discuss an important equivalence relation
between foliations, cobordant foliations.
Definition 2 Let M0 and M1 be two closed, oriented m-manifolds with codimension-q foliations. Then these fo-liated manifolds are said to be foliated cobordant if thereis a compact, oriented (m + 1)-manifold with boundary∂W = M0 ⊔M1 and with a codimension-q foliation Ftransverse to the boundary. Every leaf Lα of the folia-tion F induces leafs Lα ∩ ∂W of foliations FM0
,FM1on
the two components of the boundary ∂W .
The resulted foliated cobordism classes [FM ] of the man-ifold M form an abelian group CFm,q(M) under disjoint
union ⊔ (inverse M , unit Sq × Sm−q, see79 §29).As an example we consider a disk D2 described
by a complex number z = x + iy with |z| ≤1 and center 0 together with a foliation by leafsLx =
z = x+ iy ∈ C| − 1
2 ≤ x ≤ 12 , |y| =
√1 − x2/2
(see Fig. 3). The intersections Lx ∩ ∂D2 of the leafswith the boundary induces a foliation of ∂D2 = S1 (the
6
Figure 3. foliation of the disk D2
leafs are only points). In this example we can also reversethis process, i.e. given a foliation of S1 by points we willobtain a foliation of the disk D2 by connecting the twopoints (x,
√1 − x2/2) and (x,−
√1 − x2/2) by a geodesic
line.As an example of a foliated cobordism we consider
the foliation of the cylinder S1 × [0, 1] by the leafsLψ =
(ψ, t) ∈ S1 × [0, 1]| 0 ≤ t ≤ 1
for 0 ≤ ψ ≤ 2π.
This foliation is obviously transverse to the boundaryand induces a foliation on every S1 of the cobordism.Now we close the cobordism by adding two disks on eachside, i.e. D2 ∪ (S1 × [0, 1]) ∪ D2 = S2 to get a 2-spherewith the corresponding foliation (see Fig. 5). This fo-liation of the 2-sphere gives the obvious foliation of the3-disk D3 (with ∂D3 = S2) like in the case of the diskabove. But this foliation of the 3-disk can be understoodas a foliated cobordism between two disks forming theS2 as the boundary of the 3-disk D3. We will later usethis example in the proof of our main theorem.
B. Non-cobordant foliations of S3 detected by theGodbillon-Vey class
In81 Thurston constructed a foliation of the 3-sphereS3 which depends on a polygon P in the hyperbolic planeH2 so that two foliations are non-cobordant if the corre-sponding polygons have different areas. For later usage,we will present this construction now (see also the book79
chapter VIII for the details).Consider the hyperbolic plane H2 and its unit tangent
bundle T1H2 , i.e the tangent bundle TH2 where every
vector in the fiber has norm 1. Thus the bundle T1H2
is a S1-bundle over H2. There is a foliation F of T1H2
invariant under the isometries of H2 which is induced bybundle structure and by a family of parallel geodesics onH2. The foliation F is transverse to the fibers of T1H
2.Let P be any convex polygon in H2. We will constructa foliation FP of the three-sphere S3 depending on P .Let the sides of P be labeled s1, . . . , sk and let the angleshave magnitudes α1, . . . , αk. Let Q be the closed regionbounded by P∪P ′, where P ′ is the reflection of P throughs1. Let Qǫ, be Q minus an open ǫ-disk about each vertex.If π : T1H
2 → H2 is the projection of the bundle T1H2,
then π−1(Q) is a solid torus Q × S1(with edges) withfoliation F1 induced from F . For each i, there is anunique orientation-preserving isometry of H2, denoted Ii,which matches si point-for-point with its reflected image
s′i. We glue the cylinder π−1(si ∩ Qǫ) to the cylinderπ−1(s′i ∩ Qǫ) by the differential dIi for each i > 1, toobtain a manifold M = (S2 \ k punctures)× S1, and a(glued) foliation F2, induced from F1. To get a completeS3, we have to glue-in k solid tori for the k S1×punctures.Now we choose a linear foliation of the solid torus withslope αk/π (Reeb foliation). Finally we obtain a smoothcodimension-1 foliation FP of the 3-sphere S3 dependingon the polygon P .
Definition 3 The codimension-1 foliation F of the 3-sphere depending on a polygon P in H2 and constructedabove is called a foliation of Thurston-type.
Now we consider two codimension-1 foliations F1,F2
depending on the convex polygons P1 and P2 in H2. Asmentioned above, these foliations F1,F2 are defined bytwo one-forms ω1 and ω2 with dωa∧ωa = 0 and a = 0, 1.Now we define the one-forms θa as the solution of theequation
dωa = −θa ∧ ωa (4)
and consider the closed 3-form
ΓFa= θa ∧ dθa (5)
associated to the foliation Fa. As discovered by Godbil-lon and Vey38, ΓF depends only on the foliation F andnot on the realization via ω, θ. Thus ΓF , the Godbillon-Vey class, is an invariant of the foliation. Let F1 andF2 be two cobordant foliations then ΓF1
= ΓF2. To see
this, we consider the two 1-forms θ1, θ2 as induced fromthe corresponding 1-form θ of the foliated cobordism W
with foliation F and ∂W = S31 ∪ S3
2 (let ιa : S3a → W
be the embeddings of the two 3-spheres then ι∗aθ = θa).The Godbillon-Vey class on ∂W is given by ΓF = θ∧dθ.Now we consider the integral
GV (W,F) = 〈Γ, [W ]〉 =
∫
W
θ ∧ dθ
known as Godbillon-Vey number. Then the difference ofGodbillon-Vey numbers
GV (S31 ,F1) −GV (S3
2 ,F2) =
∫
S3
1
θ1 ∧ dθ1 −∫
S3
1
θ1 ∧ dθ1
=
∫
∂W
θ ∧ dθ
=
∫
W
d(θ ∧ dθ) = 0
vanishes, i.e. the Godbillon-Vey number (as well as theGodbillon-Vey class) is a foliated cobordism invariant.
As an example we consider the Reeb foliation FReeb onthe solid torus D2 × S1 again as defined by the 1-form ωsee (2). From the relation (3) we obtain the exact form
θ = −d(
log
(h(r) +
(1 + h(r))dfdr
))
7
as the solution of equation (4). Obviously we obtain forthe Godbillon-Vey class of the Reeb foliation
ΓFReeb= θ ∧ dθ = 0 (6)
i.e. the Godbillon-Vey class vanishes for a Reeb foli-ation. In case of the Thurston-type foliations F1,F2,Thurston81 obtains for the Godbillon-Vey number
GV (S3,Fa) =
∫
S3
ΓFa= vol(π−1(Q)) = 4π ·Area(Pa)
(7)where only the foliation on M = (S2 \ k punctures) ×S1 contributes to the Godbillon-Vey class because theReeb foliations have vanishing Godbillon-Vey class. Let[1] ∈ H3(S3,R) be the dual of the fundamental class [S3]defined by the volume form, then the Godbillon-Vey classcan be represented by
ΓFa= 4π ·Area(Pa)[1] (8)
Furthermore the Godbillon-Vey number GV seen as lin-ear functional defines a surjective homomorphism
GV : CF3,1(S3) → R (9)
from the group of foliated cobordisms CF3,1(S3) of the3-sphere to the real numbers of possible areas Area(P ),i.e.
• F1 is cobordant to F2 =⇒ Area(P1) = Area(P2)(the reverse direction (injectivity) is open)
• F1 and F2 are non-cobordant ⇐⇒ Area(P1) 6=Area(P2)
We note that Area(P ) = (k − 2)π − ∑k αk. The
Godbillon-Vey class is an element of the deRham coho-mology H3(S3,R) which will be used later to constructa relation to gerbes. Furthermore we remark that theclassification is not complete. Thurston constructed onlya surjective homomorphism from the group of cobordismclasses of foliation of S3 into the real numbers R. Weremark the close connection between the Godbillon-Veyclass (5) and the Chern-Simons form if θ can be inter-preted as connection of a suitable line bundle.
C. Codimension-one foliations on 3-manifolds
Now we will discuss the general case of a compact 3-manifold. Later on we will need the codimension-1 foli-ations of a homology 3-sphere Σ. But here we will provemore: Every codimension-1 foliation of Thurston-type ona compact 3-manifold is uniquely given by a codimension-1 Thurston-type foliation on S3.
Now we explain a procedure, the surgery along a link,to produce any compact 3-manifold Σ from a 3-sphereS3 by a link L. Let Σ be a compact 3-manifold withoutboundary. A smooth embedding L : S1⊔· · ·⊔S1 → S3 ofthe disjoint union of k circles into the 3-sphere is calleda link L with k components.
Definition 4 Let L be a link with k components in the3-sphere S3. Remove k disjoint solid tori Tk (with Tk =⊔kD
2×S1) from S3 to get S3 \Tk with boundary ∂Tk =⊔k S
1 × S1. Using a diffeomorphism φ : ∂Tk → L × S!,one can glue in L×D2 to produce a compact 3-manifold
Σ = (S3 \ Tk) ∪φ (L×D2)
We call this procedure, surgery along a link.
In55 it was proved that every compact 3-manifold with-out boundary can be obtained by using surgery along asuitable chosen link. Furthermore, Lickorish character-izes also the diffeomorphism φ. Every diffeomorphism ofa torus T 2 = S1 × S1 is given by a sequence of coordi-nate transformations (small diffeomorphisms) and Dehntwists. A Dehn twist of T 2 can be easily understood byusing the fact that the torus is given by T 2 = R2/Z2
i.e. the identification of opposite sites of a square. Soevery map Z2 → Z2 of the lattice to itself generates ahomeomorphism of the torus which is actually a diffeo-morphism (see Moise59). These maps of the lattice forma group SL(2,Z) and the Dehn twists are the generatorsof this group. Let (µ, λ) be a basis for the first homol-ogy group H1(T 2;Z). For surgery along a link, all thatis needed is the homology class α in H1(T 2;Z) which ismapped from the basis by using the diffeomorphism φof T 2,α = φ∗(µ). This class is then given uniquely byrelatively prime integers, α = pµ + qλ. Reversing theorientation of α reverses the signs of both p and q butdoesn’t affect the diffeomorphism type obtained by thesurgery, so all we need is the quotient p/q to character-ize the diffeomorphism φ. The whole procedure is calledsurgery along a link with coefficient p
q∈ Q ∪ ∞. But
the Dehn twist can be also described quite explicitly. Let(θ, φ) be two angle coordinates (range [0, 2π)), for eachS1 factor of the torus. If u(α) is a smooth function equalto one near π but zero elsewhere, we represent the (p, q)twist by
(θ, φ) → (θ + p2πu(φ), φ+ q2πu(θ)).
Or, we can define the (p, q) ∈ Z2 twisted torus asidentification space resulting from identifying (x, y) ∼(mpx, nqy) for any (m,n) ∈ Z2and (x, y) ∈ R2. For com-pleteness we state the theorem of Lickorish55 (with someadditional content74 or42 for the modern approach).
Theorem 3 (Lickorish, 196255) Every compact 3-manifold without boundary can be obtained from a 3-sphere by surgery along some link with integer coeffi-cients.
Importantly, this link L is not unique, i.e. two differentlinks can produce diffeomorphic 3-manifolds. For exam-ple, the left-handed trefoil knot (1-component link) pro-duces via surgery with coefficient −1 the Poincare sphere(the simplest homology 3-sphere) which can be also ob-tained from a surgery along the Borromean rings (linkwith 3 components) with coefficient +1 (see Rolfsen74
8
§9). There is a procedure to get from one link to an-other by using a finite number of moves, called Kirbycalculus31,49. By using this surgery along a link, weare able to construct the codimension-one foliation ofThurston-type for every compact 3-manifold.
Theorem 4 Let Σ be a compact 3-manifold withoutboundary. Every codimension-one foliation F of the 3-sphere S3 of Thurston type induces a codimension-one fo-liation FΣ of Thurston type on Σ. Every cobordism class[F ] is represented by the Godbillon-Vey invariant ΓF aselement of the deRham cohomology H3(S3,R). Thenthere exists an element in H3(Σ,R) represented by theGodbillon-Vey invariant ΓFΣ
of a cobordism class [FΣ]so that the Godbillon-Vey numbers agree GV (S3,F) =GV (Σ,FΣ).
Proof: In the construction of the Thurston-type foliationof the 3-sphere, one starts with a polygon P in H2 withk sides and constructs a foliation depending on P . In anintermediate step in this construction, one has the man-ifold M = (S2 \ k punctures) × S1 with a foliation F .To complete M to a 3-sphere S3 one has to sew in ksolid tori, each equipped with a Reeb foliation. There-fore M is S3 \ Tk in definition 4 of the surgery. Let Lbe a link with k components (arbitrary but fixed) so thatsurgery along L (with fixed coefficients determing the dif-feomorphism φ) of the 3-sphere produces the 3-manifoldΣ. Now we can sew in L×D2 via the diffeomorphism φ(defined above) representing the coefficients. We intro-duced a Reeb foliation (with suitable slope, see sectionIII B) on every (knotted) solid tori of L ×D2. Then weobtain for
Σ = M ∪φ (L×D2)
a Thurston-type foliation, or we obtain a 3-manifold Σwith foliation FΣ induced from the foliation F in S3.The cobordism class [F ] is represented by the Godbillon-Vey class ΓF in H3(S3,R) with representation ΓF =4π · Area(P )[1] ∈ H3(S3,R) (see (8)). As argued byThurston81, the Godbillon-Vey class ΓF of the foliationF does not depend on the size of the removed ǫ−disk ofeach vertex in the polygon Qǫ. Or, the Godbillon-Vey in-variant of all Reeb foliations (for the solid tori in L×D2)vanishes (see equation (6)). Therefore the surgery proce-dure does not influence the invariant ΓFΣ
of the foliationFΣ, i.e. ΓF±
= 4π ·Area(P )[1] ∈ H3(Σ,R) with the dual[1] of the volume form of Σ. After the integration weobtain for the Godbillon-Vey numbers
GV (S3,F) =
∫
S3
ΓF = 4π·Area(P ) =
∫
Σ
ΓFΣ= GV (Σ,FΣ)
Thus the value of the invariant of the foliation F is equalto the value of the invariant of the foliation FΣ.
IV. EXOTIC R4 AND CODIMENSION-ONEFOLIATIONS
In this section we will describe the construction ofsmall exotic R4’s by using the failure of the smooth hcobordism theorem for some compact 4-manifolds. Thestructure theorem for smooth h cobordisms of compact4-manifolds28 singles the Akbulut cork out as the reasonfor this failure. This description has the great advantageto have an explicit coordinate representation. The mainpart of this representation is given by an infinite con-struction typically for 4-manifolds, the Casson handle.Here we can only present the tip of the iceberg. Thereare very good books like the classical ones32,51 or modernviews like42 as well the original research articles27,33–35,73
to understand Casson handles, capped gropes and its de-sign. Using this machinery, DeMichelis and Freedman29
were able to construct a family of uncountable, manynon-diffeomorphic exotic R4 (a radial family). Finally insubsection IV E, we use this radial family to get a directrelation (Theorem 5) to non-cobordant, codimension-1foliations of the boundary ∂A of the Akbulut cork A.
A. Exotic R4 and Casson handles
The theorem 2 relates a non-trivial h-cobordismbetween two compact, simply-connected, smooth 4-manifolds to a small exotic R4. Using theorem 1, we canunderstand where the non-triviality of the h-cobordismcomes from: one of the Akbulut corks, say A, mustbe glued in by using a non-trivial involution of theboundary ∂A. In the notation above, there is a non-product h-cobordism W between M4 and M4
0 with a h-subcobordism X between A0 ⊂M0 and A ⊂M . There isan open neighborhood U of the h-subcobordism X whichis an open h-cobordism U between the open neighbor-hoods N(A) ⊂ M, N(A0) ⊂ M0. Both neighborhoodsare homeomorphic to R4 but not diffeomorphic to eachother (as induced from the non-productness of the h-cobordism W ).
Let us consider now the basic construction of the Cas-son handle CH . Let M be a smooth, compact, simply-connected 4-manifold and f : D2 →M a (codimension-2)mapping. By using diffeomorphisms of D2 and M , onecan deform the mapping f to get an immersion (i.e. injec-tive differential) generically with only double points (i.e.#|f−1(f(x))| = 2) as singularities39. But to incorporatethe generic location of the disk, one is rather interest-ing in the mapping of a 2-handle D2 × D2 induced byf × id : D2 ×D2 →M from f . Then every double point(or self-intersection) of f(D2) leads to self-plumbings ofthe 2-handle D2 × D2. A self-plumbing is an identifi-cation of D2
0 × D2 with D21 × D2 where D2
0 , D21 ⊂ D2
are disjoint sub-disks of the first factor disk68. Considerthe pair (D2×D2, ∂D2×D2) and produce finitely manyself-plumbings away from the attaching region ∂D2×D2
to get a kinky handle (k, ∂−k) where ∂−k denotes the
9
attaching region of the kinky handle. A kinky handle(k, ∂−k) is a one-stage tower (T1, ∂
−T1) and an (n+ 1)-stage tower (Tn+1, ∂
−Tn+1) is an n-stage tower unionkinky handles
⋃nℓ=1(Tℓ, ∂
−Tℓ) where two towers are at-tached along ∂−Tℓ. Let T−
n be (interiorTn) ∪ ∂−Tn andthe Casson handle
CH =⋃
ℓ=0
T−
ℓ
is the union of towers (with direct limit topology inducedfrom the inclusions Tn → Tn+1). A Casson handle isspecified up to (orientation preserving) diffeomorphism(of pairs) by a labeled finitely-branching tree with base-point *, having all edge paths infinitely extendable awayfrom *. Each edge should be given a label + or −. Hereis the construction: tree → CH . Each vertex corre-sponds to a kinky handle; the self-plumbing number ofthat kinky handle equals the number of branches leavingthe vertex. The sign on each branch corresponds to thesign of the associated self plumbing. The whole processgenerates a tree with infinite many levels. In principle,every tree with a finite number of branches per level re-alizes a corresponding Casson handle. The simplest non-trivial Casson handle is represented by the tree Tree+:each level has one branching point with positive sign +.
Given a labeled based tree Q, let us describe a subsetUQ of D2×D2. Now we will construct a (UQ, ∂D
2×D2)which is diffeomorphic to the Casson handle associatedto Q. In D2×D2 embed a ramified Whitehead link withone Whitehead link component for every edge labeledby + leaving * and one mirror image Whitehead linkcomponent for every edge labeled by −(minus) leaving*. Corresponding to each first level node of Q we havealready found a (normally framed) solid torus embeddedin D2×∂D2. In each of these solid tori embed a ramifiedWhitehead link, ramified according to the number of +and − labeled branches leaving that node. We can dothat process for every level of Q. Let the disjoint unionof the (closed) solid tori in the nth family (one solid torusfor each branch at level n in Q) be denoted by Xn. Qtells us how to construct an infinite chain of inclusions:
. . . ⊂ Xn+1 ⊂ Xn ⊂ Xn−1 ⊂ . . . ⊂ X1 ⊂ D2 × ∂D2
and we define the Whitehead decomposition WhQ =⋂∞
n=1Xn of Q. WhQ is the Whitehead continuum82 forthe simplest unbranched tree. We define UQ to be
UQ = D2 ×D2 \ (D2 × ∂D2 ∪ closure(WhQ))
alternatively one can also write
UQ = D2 ×D2 \ cone(WhQ) (10)
where cone() is the cone of a space
cone(A) = A× [0, 1]/(x, 0) ∼ (x′, 0) ∀x, x′ ∈ A
over the point (0, 0) ∈ D2 × D2. As Freedman (see34
Theorem 2.2) showed UQ is diffeomorphic to the Cassonhandle CHQ given by the tree Q.
B. The design of a Casson handle and its foliation
A Casson handle is represented by a labeled finitely-branching tree Q with base point ⋆, having all edge pathsinfinitely extendable away from ⋆. Each edge shouldbe given a label + or − and each vertex correspondsto a kinky handle where the self-plumbing number ofthat kinky handle equals the number of branches leav-ing the vertex. The open handle D2 × R2 is repre-sented by the ⋆, i.e. there are no kinky handles. Oneof the cornerstones of Freedmans proof of the homeo-morphism between a Casson handle CH and the open2-handle H = D2 × R2 are the reembedding theorems.Then one foliates CH and H by copies of the frontierFr(CH). The frontier of a set K is defined by Fr(K) =closure(closure(K) \ K). As example we consider theinterior int(D2) of a disk and obtain for the frontierFr(int(D2)) = closure(closure(int(D2)) \ int(D2)) =∂D2, i.e. the boundary of the disk D2. Then Freedman(34 p.398) constructs another labeled tree S(Q) from thetree Q. There is a base point from which a single edge(called “decimal point”) emerges. The tree is binary: oneedge enters and two edges leaving a vertex. The edges arenamed by initial segments of infinite base 3-decimals rep-resenting numbers in the standard “middle third” Cantorset69 CS ⊂ [0, 1]. Each edge e of S(Q) carries a label τewhere τe is an ordered finite disjoint union of 5-stagetowers together with an ordered collection of standardloops generating the fundamental group. There is threeconstraints on the labels which leads to the correspon-dence between the ± labeled tree Q and the (associated)τ -labeled tree S(Q). One calls S(Q) the design.
Two words are in order for the design S(Q): first, everysequence of 0’s and 2’s is one path in S(Q) representingone embedded Casson handle CHQ1
⊂ CHQ where bothtrees are related like Q ⊂ Q1. For example, the Cas-son handle corresponding to .020202... is obtained as theunion of the 5-stage towers T 0 ∪ T 02 ∪ T 020 ∪ T 0202 ∪T 02020 ∪ T 020202 ∪ .... For later usage we identify the se-quence .00000...with the Tree Tree+. Secondly, there aregaps, i.e. we have only a Cantor set of Casson handlesnot a continuum. For instance a gap is lying between thepaths .022222 . . . and .20000 . . . In the proof of Freed-man, the gaps are shrunk to a point and one gets thedesired homeomorphism. But the gaps are also impor-tant to understand the design as well its frontier. Letγ ∈ S(Q) be a path and Fγ the frontier with its closureS1×D2/Whγ (see the previous subsection). The interiorof every Casson handle is diffeomorphic to the standardR4. Therefore the frontier Fγ is most important to un-derstand the Casson handle. Now consider the two pathsγ = .022222 . . . and γ′ = .20000 . . .. Every path in S(Q)is represented by one sequence over the alphabet 0,2.Every gap is a sequence containing at least one 1 (so forinstance .1222... or .012222...). In fact, the gap betweenγ and γ′ corresponds to the first middle third which isdeleted by constructing S(Q) and one has the relationγ ∪ γ′ = ∂[.1n1n2 . . .] = ∂[all numbers beginning .1] (see
10
Figure 4. structure of the design S(Q) as 5-stage tower struc-ture (fig. repainted from51)
chapter XIII §4 of51). Therefore the boundaries of thegaps are the important part to understand the design.
Here we will use this structure to produce a foliation ofthe design. There is now a natural order structure givenby the sequence (for instance .022222... < .12222... <.22222...). The leaves are the corresponding gaps or Cas-son handles (represented by the union 5-stage towers end-ing with T 02222..., T 12222.. or T 22222...). The tree struc-ture of the design S(Q) should be also reflected in thefoliation to represent every path in S(Q) as a union of 5-stage towers. By the reembedding theorems, the 5-stagetowers can be embedded into each other (see Fig. 4)
Then we obtain two foliations of the (topological)open 2-handle D2 × R2: a codimension-1 foliationalong one R−axis labeled by the sequences (for in-stance .022222... < .12222... < .22222...) and a secondcodimension-1 foliation along the radius of the disk D2
induced by inclusion of the 5-stage towers (for instanceT 0 ⊃ T 02 ⊃ T 020 ⊃ ...). Especially the exploration of aCasson handle by using the design is given by its fron-tier, in this case, minus the attaching region. In caseof a usual tower we get the frontier S1 ×D2/Whγ withγ ∈ S(Q). The gaps have a similar structure. Then thefoliation of the Casson handle (induced from the design)is given by the leaves S1 × D1 over the disk D2 in theCasson handle, i.e. the disk D2 is foliated by parallel
Figure 5. foliation of the 2-sphere as foliated cobordism E ofthe two disks N,S
lines (see Fig. 3). So, every Casson handle with a giventree Q has a codimension-one foliation given by its de-sign. This foliation can be also understood as a foliatedcobordism.
Lemma 1 The foliation of the design is induced by afoliation of the gaps seen as foliated cobordism.
Proof: Gaps are characterized as sequences containing a1, so that we have a countable number of gaps. Abovewe discussed the relation
γ ∪ γ′ = ∂[.1n1n2 . . .] = ∂[all numbers beginning .1]
between the two paths γ = .022222 . . . and γ′ =.20000 . . . and the gap .1n1n2 . . . . Gaps in the Cassonhandle look like S1×D3/Wh. Therefore we have to con-centrate on S1×D3 and its boundary (or better its fron-tier) S1 × S2 (or better S1 × S2/Wh) to understand thefoliation of the whole design. For that purpose we con-sider the foliation as part of a foliation of the 2-sphere(see Fig. 5) like in the example at the end of sectionIII A. The 2-sphere is decomposed by S2 = N ∪ E ∪ S,two pole regions N,S (N,S = D2) and an equator re-gion E = S1 × D1. The foliation of the disk as in Fig.3 can be used to foliate N and S. Both foliations canbe connected by the leaves S1 which are the longitudes.Then one obtains a foliated cobordism D3 (see the ex-amples at the end of subsection III A) between N and Sgiven by the obvious foliation of the equator region E (acylinder).
C. Capped gropes and its design
In this subsection we discuss a possible generalizationof Casson handles. The modern way to the classification
11
Figure 6. Example of a grope with symplectic basis as curvesaround the holes
of 4-manifolds used “capped gropes”, a mixed variant ofCasson handle and grope (chapters 1 to 4 in32). We donot want to complicate the situation more than needed.But for later developments we have to discuss some partof the theory but we remark that all results can be easilygeneralized to capped gropes as well.
A grope is a special pair (2-complex,circle), where thecircle is referred to as the boundary of the grope. Thereis an anomalous case when the depth is 1: the uniquegrope of depth 1 is the pair (circle,circle). A grope ofdepth 2 is a punctured surface with the boundary circlespecified (see Fig. 6). To form a grope G of depth n,take a punctured surface, F , and prescribe a symplecticbasis αi, βj. That is, αi and βj are embedded curvesin F which represent a basis of H1(F ) such that the onlyintersections among the αi and βj occur when αi and βjmeet in a single point αi · βj = 1. Now glue gropes ofdepth < n along their boundary circles to each αi andβj with at least one such added grope being of depthn − 1. (Note that we are allowing any added grope tobe of depth 1, in which case we are not really adding agrope.) The surface F ⊂ G is called the bottom stage ofthe grope and its boundary is the boundary of the grope.The tips of the grope are those symplectic basis elementsof the various punctured surfaces of the grope which donot have gropes of depth > 1 attached to them.
Definition 5 A capped grope is a grope with disks (thecaps) attached to all its tips. The grope without the capsis sometimes called the body of the capped grope.
The capped grope (as cope) was firstly described byFreedman in 198335. The caps are only immersed diskslike in case of the Casson handle to make the gropesimply-connected. The great advantage is the simplerfrontier, instead of S1 ×D2/Whγ (see subsection IV B)one has solid tori S1 × D2 as frontier of the cappedgrope (as shown in6). The corresponding design (andits parametrization) can be described similar to the Cas-son handle by sequences containing 0 and 2 (see section
4.5 in32). There are also gaps (described by a 1 in thesequence) who look like S1×D3. Because of the simplierfrontier, the lemma 1 can be easily generalized to cappedgropes.
D. The radial family of uncountably infinite small exoticR4
Let a small exotic R4, ,be induced from the non-product h-cobordism W between M and M0 with Ak-bulut corks A ⊂ M and A0 ⊂ M0, respectively. LetK ⊂ R4 be a compact subset. Bizaca and Gompf20
constructed the small exotic R41 by using the simplest
tree Tree+. Bizaca18,19 showed that the Casson han-dle generated by Tree+ is an exotic Casson handle.Using Theorem 3.2 of29, there is a topological radiusfunction ρ : R4
1 → [0,+∞) (polar coordinates) so thatR4t = ρ−1([0, r)) with t = 1 − 1
r. Then K ⊂ R4
0 and R4t
is also a small exotic R4 for t belonging to a Cantor setCS ⊂ [0, 1]. Especially two exotic R4
s and R4t are non-
diffeomorphic for s < t except for countable many pairs.In29 it was claimed that there is a smoothly embeddedhomology 4-disk A. The boundary ∂A is a homology3-sphere with a non-trivial representation of its funda-mental group into SO(3) (so ∂A cannot be diffeomorphicto a 3-sphere). According to Theorem 2 this homology4-disk must be identified with the Akbulut cork of thenon-trivial h-cobordism. The cork A is contractable andcan be (at least) build by one 1-handle and one 2-handle(case of a Mazur manifold). Given a radial family R4
t
with radius r = 11−t so that t = 1 − 1
r⊂ CS ⊂ [0, 1].
Suppose there is a diffeomorphism
(d, idK) : (R4s,K) → (R4
t ,K) s 6= t ∈ CS
fixing the compact subset K. Then this map d in-duces end-periodic manifolds70 M \ (
⋂∞
i=0 di(R4
s)) andM0 \ (
⋂∞
i=0 di(R4
s)) which must be smoothable contra-dicting a theorem of Taubes80. Therefore R4
s and R4t are
non-diffeomorphic for t 6= s (except for countable manypossibilities).
E. Exotic R4 and codimension-1 foliations
In this subsection we will construct a codimension-onefoliation on the boundary ∂A of the cork with non-trivialGodbillon-Vey invariant. The strategy of the proof goeslike this: we use the foliation of the design of the Cassonhandle (see subsection IV B) for the radial family R4
t toinduce a foliated cobordism ∂A × [0, 1]. The restrictionto its boundary gives cobordant codimension-1 foliationsof ∂A with non-trivial Godbillon number r2 = 1
(1−t)2 .
Theorem 5 Given a radial family Rt of small exotic R4t
with radius r and t = 1 − 1r⊂ CS ⊂ [0, 1] induced from
the non-product h-cobordism W between M and M0 withAkbulut cork A ⊂ M and A ⊂ M0, respectively. The
12
Figure 7. decomposition of the boundary of a gap (fig. re-painted from29)
radial family Rt determines a family of codimension-onefoliations of ∂A with Godbillon-Vey number r2. Further-more given two exotic spaces Rt and Rs, homeomorphicbut non-diffeomorphic to each other (and so t 6= s) thenthe two corresponding codimension-one foliation of ∂Aare non-cobordant to each other.
Proof: We consider a tubular neighborhood ∂A× [0, 1] ⊂R4
1 of ∂A and glue the Casson handle along some 2-handle. Now we will weaken the Casson handles by us-ing capped gropes (see subsection IV C and chapters 1-4in32) denoted by GCH. These differ from Casson handlesin that many surface stages are interspersed between theimmersed disks of Casson’s construction. The GCH arealso indexed by rooted finitely branching objects. Thegrowth rate of their stages was determined in6 (TheoremA) to be at least exponential (more than 2n). In theproof of Theorem 3.2 in29 the gaps in the design whereused. The gaps in case of the Casson handle are not man-ifolds and look like S1 ×D3/Wh. In case of the cappedgrope one has “good” gaps of the form S1 × D3. Thatis the reason why we switch to these objects now. Nowwe decompose the gap by gap = S1 ×D3 = S1 ×D2 × Iwith the unit interval I = [0, 1] = D1. The boundary is adecomposition ∂(gap) = (S1×S1×I)∪(S1×D2×0, 1)of the caps (north and south) and the equator region (seeFig. 7). We remark the importance of the boundary ofthe gap (see the proof of lemma 1).
The radius coordinate ρ defined above is identifiedwith the unit interval of the gap (see the proof of The-orem 3.2). In the notation of29, we think of each gapas gap = S1 × N × I where N = D2 is the neighbor-hood around the north pole of the 2-sphere in Fig. 7.Using the reembedding theorems every GCH embeds inthe open 2-handle and induces a foliation visualized inFig. As described in subsection IV B the simplest treeTree+ belongs to the binary sequence .000 . . . and is rep-resented by t = 1 and the radius r = 1/(1 − t) = +∞.The foliation of the design is perpendicular to S1 × N ,i.e. S1 × lattitude are the leaves. The intersectionof the leaves with S1 × N produces a foliation of thedisk N as proved in theorem 1. This disk is given up toconformal automorphism by fixing the sphere S2 ⊃ N ,i.e. the disk is invariant w.r.t. the group PSL(2,R).The boundary of N is given by geodesic curves. The
Figure 8. visualization of foliation as coordinatization of thedesign (fig. repainted from29)
PSL(2,R)−invariance induces a mapping of the disk Ninto the hyperbolic space H2, where every PSL(2,R)transformation is an isometry now. Then the foliated Nis mapped to a foliated polygon P in H2, where the fo-liation is PSL(2,R)−invariant. From this point of viewwe interpret S1 × N as the unit tangent bundle of thepolygon T1P . Then the volume of the polygon P is thevolume of the disk N , i.e. vol(P ) = vol(N) and wechoose the number of vertices of P in a suitable man-ner by defining the geodesic arcs forming the boundaryof N . As Fig. 7 indicated, the disk N is also part of theboundary ∂(gap) = S1×S2 of the gap using its foliation,see lemma 1. Then the unit interval in the gap is di-rectly related to the radius r of the 2-sphere S2 ⊃ N andthis radius determines the volume of the disk N (as partof the upper hemisphere of S2, see Fig. 7). But thenby using PSL(2,R)−invariance, we obtain the relationvol(P ) = r2.
The tubular neighborhood ∂A × [0, 1] = ∂A × I canbe chosen in such a manner that the coordinate ρ agreeswith the unit interval I of the neighborhood. In sub-section IV B we described the foliation of the design as afoliated cobordism between two disks N,S given by the 2-sphere S2 (see Fig. 5 and lemma 1). As described above,every disk (N or S) is related to the polygon P (withoutloss of generality we use N and S with the same volumevol(N) = vol(S)). Then the foliation of the design in-duces a foliation of the cobordism ∂A× I. The foliationof the 3-disk D3 can be seen as the foliated cobordism be-tween two disks (see the example at the end of subsectionIII A). The space D2×S2 (i.e. the 5-stage towers) is theleaf of the foliated cobordism transverse to the foliationon the boundary ∂A. Then the restriction on the bound-ary ∂A × 0, 1 induces a foliation on ∂A determinedby the volume vol(P ) of the polygon. So, the radius r2
(proportional to the volume of vol(P )) is a cobordisminvariant of the foliation. By Theorem 4 we obtain acodimension-1 foliation of ∂A induced from a foliation ofthe S3. As shown81 (see also the book79 chapter VIII forthe details) this invariant agrees with the Godbillon-Veynumber GV = r2. Then two non-diffeomorphic smallexotic R4 (for s 6= t) have different radial coordinates( 11−s = rs 6= rt = 1
1−t ) and therefore different Godbillon-
Vey invariants r2s 6= r2t . The corresponding foliations are
13
non-cobordant to each other.
In the theorem 5 above we constructed a relation be-tween codimension-1 foliations on Σ = ∂A and the radiusfor the radial family of small exotic R4. By using theo-rem 4 we can trace back the foliation on Σ by a foliationon the 3-sphere S3. This situation can be seen differentlyby using the property of the exotic R4 to be small, i.e.there is a smoothly embedded 3-sphere S3. The foliationon Σ = ∂A induces a foliation of the whole cork A. Thus,if there is a smoothly embedded 3-sphere S3 lying insidethe Akbulut cork A, then the a codimension-1 foliationon Σ induces a codimension-1 foliation on the embedded3-sphere. The 3-sphere can be made large enough that itlies in the tubular neighborhood ∂A× [0, 1]. We will saythat S3 lies at the boundary ∂A = Σ of the cork A. Thenby theorem 5:
Corollary 1 Any class in H3(S3,R) induces a small ex-otic R4 where S3 lies at the boundary Σ = ∂A of the corkA.
F. Integer Godbillon-Vey invariants and flat bundles
Clearly the integer classes H3(S3,Z) ⊂ H3(S3,R) area subset of the full set and one can use the construc-tion above to get the foliation. Especially the polygonP must be formed by segments with angles αk of in-teger value with respect to π to get an integer valuefor the volume Area(P ) = (k − 2)π −∑k αk up to aπ−factor. Using the work of Goldman and Brooks25, onecan construct a foliation admitting an integer Godbillon-Vey invariant. The corresponding foliation is inducedby the unit tangent T1H
2 or by the action of the Mbiusgroup PSL(2,R) = SL(2,R)/Z2 (Remark: PSL(2,R)acts transitively on T1H
2 and so we can identify bothspaces). The unit tangent bundle T1H
2 = PSL(2,R) is acircle bundle over H2 and we can construct the universalcover, a real line bundle over H2, denoted by SL(2,R). Insubsection III B we described the Thurston constructionof a codimension-1 foliation F . In an intermediate stepone has the manifoldM = (S2\k punctures)×S1 (witha foliation F). This foliation F is defined by a one-formω together with two other 1-forms θ, η with
dω = θ ∧ ω, dθ = ω ∧ η, dη = η ∧ θ (11)
and Godbillon-Vey invariant GV (F) = θ∧dθ = ω∧η∧θ.Now we show that the Godbillon-Vey invariant of thisfoliation F is an integer 3-form:
Lemma 2 Let M be a 3-manifold with non-trivial fun-damental group π1(M) and a foliation F defined bythe 1-form ω together with two 1-forms θ, η fulfillingthe relations (11). If M can be written as a flatPSL(2,R)−bundle over a manifold N with fiber S1 andπ1(N) 6= 0. Then the pairing of the Godbillon-Vey in-variant ΓF with the fundamental class [M ] ∈ H3(M) is
given by
GV (M,F) = 〈ΓF , [M ]〉 =
∫
M
ΓF = ±(4π)2 · χ(N) (12)
with the Euler characteristic χ(N) of N . Up to a nor-malization constant and the orientation one obtains aninteger value.
Proof: According to the Thurston’s construction, wechoose M = (S2 \ k punctures) × S1 and N = S2 \k punctures. The group PSL(2,R) is the isometrygroup of the hyperbolic plane H2 leaving the foliationof M invariant. Then (see25) the holonomy of the foli-
ation is given by a homomorphism π1(M) → SL(2,R)defining a flat bundle over N . Using Proposition 2 of25,the integration of the Godbillon-Vey invariant over thefiber S1 gives
∫
S1
GV (F) = (4π)2 · e(M)
with the Euler class of the flat bundle M . An integrationover N gives by definition the Euler characteristics χ(N).Thus we obtain the desired result:
GV (M,F) = 〈ΓF , [M ]〉 =
∫
M
ΓF = (4π)2 · χ(N)
the integer invariant. A change of the orientation changesthe sign of the integral. Then we obtain the negativeintegers.
Using this lemma we are able to obtain the Thurstontype foliation of the S3 with integer Godbillon-Vey in-variant.
Theorem 6 Every PSL(2,R) flat bundle over M =(S2 \ k punctures) × S1 defines a codimension-1 foli-ation of M by the horizontal distribution of the flat con-nection so that its (normalized) Godbillon-Vey invariantis an integer given by
1
(4π)2〈ΓF , [M ]〉 = ±χ(N) = ±(2 − k) . (13)
This foliation can be extended to the whole 3-sphere S3
defining an integer class in H3(S3,Z).
Proof: The first half of the theorem is trivial using thelemma 2. The Euler characteristic χ(N) is given bythe Euler characteristic of the 2-sphere χ(S2) minus thenumber k of punctures. Then in the construction ofThurston, one sew in k Reeb components having van-ishing Godbillon-Vey invariant (see equation (6)). Thenthe Godbillon-Vey invariant of the foliation of S3 is thesame as GV (F). The pairing (13) is an integer andthe form is in the image of the injective homomorphismH3(S3,Z) → H3(S3,R). Then we identify the deRhamform with the integer class in H3(S3,Z) by this homo-morphism.
14
It is an important consequence of the work25 that thefoliation F (and its induced counterpart for the 3-sphereS3) is rigid, i.e. a disturbance (or continuous variation)does not change the Godbillon-Vey invariant.
G. The conjecture: the failure of the smooth4-dimensional Poincare conjecture
In this section we will go a step further. We considerthe 4-sphere and use the close relation between foliationsand smooth structures. In case of the 4-sphere S4 weneed an exotic 4-disk D4 which is the Akbulut cork forS4. The boundary of D4 is the 3-sphere S3. Now weconsider a polygon Pk in H2 representing a codimension-1 foliation of the 3-sphere. In analogy to the constructionabove we attach a Casson handle to the S3 which is re-lated to the polygon Pk. But now we have a big differenceto the non-compact exotic R4. The 4-sphere is compactand we have to consider an embedding of the Cassonhandle. This embedding leads to the cut-off of the Cas-son handle after N levels (see75 for the argument usingPL methods). This finite numbers restricts the possiblevalues of the area Area(Pk) enough to obtain countablemany values only. Thus the integer values in H3(S3,Z)of the Godbillon-Vey invariant (see also the previous sub-section) is represented by the foliation of the 3-sphere S3
as part of the compact 4-sphere S4. Using the work ofMather56, every topological foliation (of type C0 i.e. ofHaefliger type54) on S3 can be uniquely extended to D4.But this fails for any C2 foliation where the obstructionis the Godbillon-Vey invariant. This motivates the con-jecture:
Conjecture 1 Every 3-sphere admitting a codimension-1 foliation with integer Godbillon-Vey invariant boundsan exotic 4-disk D4.
But the existence of an exotic 4-disk implies the failure ofthe smooth Poincare conjecture (SPC4)71 in dimension4. Or,
Conjecture 2 If we glue an exotic 4-disk to a standard4-disk along its common boundary we obtain an exotic 4-sphere. Thus we obtain the failure of the smooth Poincareconjecture in dimension 4.
For the following, it is interesting to note by using theconjecture that countable infinite exotic structures ofthe 4-spheres are related to the elements in H3(S3,Z)(canonically isomorphic to H4(S4,Z)). Unfortunatelythe conjecture cannot be checked by the lack of a suitableinvariant.
V. ABELIAN GERBES ON S3
In this section we will focus on some constructions re-ferring to gerbes. Abelian gerbes and gerbe bundles on
manifolds are geometrical objects interpreting third inte-gral cohomologies on manifolds, similarly as isomorphismclasses of linear complex line bundles are classified byH2(M,Z). From that point of view abelian gerbes areimportant for the constructions in this paper. There ex-ists a vast literature in mathematics and physics devotedto gerbes and bundle gerbes. Gerbes were first consid-ered by Giraud37. The classical reference is the Brylinskibook26. Here we are interested in abelian gerbes. Upto uniqueness questions of the choices we can, followingHitchin, make use of the simplified working definition ofa gerbe45. Hence we do not refer to categorical construc-tions (sheaves of categories, see e.g.60) which, from theother hand, are essential for correct recognition of gerbes.In that way we can state easily the relation of gerbes to3-rd integral cohomologies on S3. In the next section wewill comment on categorically defined 2-gerbes in contextof smooth exotic structures.
Abelian, or S1, gerbes are best understood in termsof cocycles and corresponding transition objects, like S1
principal bundles. This bundle is specified by a cocyclegαβ : Uα ∩ Uβ → S1 which is the Cech cocycle from
C1(M,C∞(S1)) where Uα,β are elements of a good coverof a manifold M . However, to define a S1 gerbe, we needto compare data on each triple intersections of elementsfor a good cover. Hence, let gαβγ : Uα ∩ Uβ ∩ Uγ → S1
be the cocycle in C2(M,C∞(S1)) with
gαβγ = g−1βαγ = g−1
βγα = g−1αγβ (14)
satisfying the cocycle condition
δg = gβγδg−1αγδgαβδg
−1αβγ = 1 (15)
on each fourth intersection Uα∩Uβ ∩Uγ ∩Uδ. The abovedata defines the S1−gerbe.
The cocycles in C2(M,C∞(S1)) give rise (up to thecoboundaries) to the cohomology group H2(M,C∞(S1))which classifies gerbes as defined above. However, wehave the canonical exact sequence of sheaves on M
0 −→ Z −→ C∞(R) −→ C∞(S1) −→ 1 (16)
where the third morphism is given by e2πix. However,C∞(R) is fine, hence
H2(M,C∞(S1)) = H3(M,Z) (17)
We see that gerbes are classified by third integral co-homologies on M similarly as line bundles are classi-fied topologically by Chern classes. The elements ofH3(M,Z) are called the Dixmier-Douady classes of the“gerbe local data”. It is worth noticing that gerbes areneither manifolds nor bundles. These can be consideredas generalization of both: sheaves (bundle gerbes) andvector bundles (cocycle description).
15
A trivialization of a S1- gerbe is given by functions
fαβ = f−1βα : Uα → Uβ (18)
such that
gαβγ = fαβfβγfγα (19)
which is a representation of a cocycle by functions. Thusthe difference of two trivializations is given by hαβ =
fαβ/f′αβ which means hαβ = h−1
βα and
hαβhβγhγα = 1 (20)
and this is exactly a cocycle for some line bundle on M .We say that the (generalized) transition functions of anabelian gerbe are line bundles. One can iterate this con-struction and define higher ,,gerbes” with the generalizedtransition functions given by lower rank gerbes.A connection on a gerbe is specified by 1-forms Aαβ
and 2-forms Bα satisfying the following two conditions
iAαβ + iAβγ + iAγα = g−1αβγdgαβγ (21)
Bβ −Bα = dAαβ (22)
This implies that there exists a globally defined 3-formH ,with integral 3-rd deRham cohomologies correspondingto [H/2π] and defined by local 2-forms Bα
H |Uα= dBα (23)
This 3-formH is the curvature of a gerbe with connectionas above. The local data defining a gerbe exists wheneverthe 3-form H has its 3-periods in 2πZ, hence [H/2π] isintegral.
The trivialization fαβ is the element of
C1(M,C∞(S1)) which implies that when a gerbeis trivial then gαβγ satisfies
[gαβγ ] = 0 (24)
The connection on a gerbe is flat when H |Uα= dBα = 0.
In that case we have Bα = daα for a suitably chosencover.
In case of M = S3 the integral third deRham co-homology group (as image of the map H3(S3,R) →H3(S3,Z)) classify S1-gerbes on S3. These integralclasses are Dixmier-Douady classes for local data of thegerbes. The canonical S1-bundle gerbe on the 3-sphereS3 was first constructed in36 and later47,57. The canoni-cal U(1) - gerbe G on S3 corresponds to the 3-form H =1
12π tr(g−1dg)3. Other gerbes on S3, correspond to the
curvatures kH, k ∈ Z, and are determined by the tensorpowers Gk. Given kH, k ∈ Z one has the unique gerbe Gkup to stable isomorphism, since H2(SU(2), U(1)) = 1.
Thus, we can uniquely represent (up to stable isomor-phisms of gerbes) the action of elements in H3(S3,Z) bythe deformations caused by the corresponding gerbes. Inparticular the abelian gerbe on S3 determines its uniqueclass as the element in H3(S3,Z)This element determinesalso an exotic R4 from the radial family which followsfrom the combination of theorems 5 and 6. Thus, re-membering that Σ = ∂A is the boundary of the Akbulutcork, we have the result:
Theorem 7 Every exotic R4 of the radial family withinteger Godbillon-Vey number of the Thurston-type foli-ation in Σ (and S3) can be represented by, and in factgenerated from, an abelian gerbe with connection on Σ(and S3). In particular, the effects of the change of thesmooth structure on R4 from the standard one to any ex-otic structure (from the radial family) having integer GVinvariant, can be described by the suitable twisting of spe-cial structures on S3, caused by the unique abelian gerbeon S3.
This connection is further studied in the next section .
VI. GENERALIZED (COMPLEX) STRUCTURES OFHITCHIN
In this section we are looking again for tools based ongerbe constructions to distinguish between small exoticR4’s corresponding to integral classes in H3(S3,R). Thegeneral non-integral case is naturally captured in this set-up by the use of generalized geometries of Hitchin. Tolook for the effects of various exoticR4’s in the geometriesof S3 was proposed by Asselmeyer and Brans9 but de-rived in Sec. IV. S3 has an unique smooth structure andthe change of smooth structures on R4 cannot affect thesmoothness of S3. However, the geometries, foliations orsome K-theoretic and categorical ingredients in dimen-sion three, can vary along with the change of the ambient4-smoothness. In this section we show the correspon-dence between structures defined on S3 whose variationsreflect the change of smoothness on R4. These structuresare generalized geometries and complex structures62 in-troduced by Hitchin45 now called Hitchin structures.
Generalized structures are based on the substitutionof the tangent space TM of a manifold M by the sumTM ⊕ T ⋆M of the tangent and cotangent bundles suchthat the spin structure for this generalized ,,tangent”bundle becomes the bundle of all forms ∧•M on M .Our interest in Hitchin’s structures is due to a corre-spondences between the deformations of these structuresand small exotic R4 expressed by the theorem 8 below.
The class of H- deformed Hitchin’s structures on S3
could be as well referred to as H-twisted Courant brack-ets on TS3 ⊕ T ⋆S3 and can be integrable with respectto the brackets. In the following we will explain main
16
points of this correspondence. An excellent reference forgeneralized geometries and complex structures is43. Theproof of Theorem 8 will be presented later. Now we willintroduce the relevant constituents.
Definition 6 Given a smooth manifold M , the Courantbracket [ , ] is defined on smooth sections of TM ⊕T ⋆M ,by
[X+ξ, Y +η] = [X,Y ]+LXη−LY ξ−1
2d(iXη−iY ξ) (25)
where X + ξ, Y + η ∈ C∞(TM ⊕ T ⋆M), LX is the Liederivative in the direction of the field X, iXη is the innerproduct of a 1-form η and a vector field X. A Courantbracket on M is also called a Hitchin structure or gener-alized geometric (complex) structure
On the RHS of (25) [ , ] is the Lie bracket on fields. Thisfact is not misleading since the Courant bracket reducesto the Lie bracket for vector fields, i.e. π([X,Y ]) =[π(X), π(Y )] where π : TM ⊕ T ⋆M → TM . It followsthat the bracket is skew symmetric and vanishes on 1-forms. However, the Courant bracket is not a Lie bracket,since it does not fulfill the Jacobi identity. The expres-sion measuring the failure of the identity is the followingJacobiator:
Jac(X,Y, Z) = [[X,Y ], Z]+[[Y, Z], X ]+[[Z,X ], Y ] (26)
The Jacobiator can be expressed as the derivative of aquantity which is Nijenhuis operator, and it holds
Jac(X,Y, Z) = dNij(X,Y, Z) (27)
Nij(X,Y, Z) =1
3(〈[X,Y ], Z〉 + 〈[Y, Z], X〉 + 〈[Z,X ], Y 〉)
(28)where 〈 , 〉 is the inner product on TM ⊕ T ⋆M . Thisinner product is a naturally given by
〈X + ξ, Y + η〉 =1
2(ξ(Y ) + η(X)) (29)
This product is symmetric and has the signature (n, n),where n = dim(M), having the non-compact orthogonalgroup O(TM ⊕ T ⋆M) = O(n, n) as symmetry of theproduct.
Definition 7 A subbundle L < TM ⊕ T ⋆M is involu-tive iff it is closed under the Courant bracket defined onits smooth sections, and is isotropic when 〈X,Y 〉 = 0for X, Y smooth sections of L. In case of dim(L) = n(maximality) we call the isotropic subbundle a maximalisotropic subbundle.
The following property characterizes this sub-bundles(see43, Proposition 3.27):If L is a maximal isotropic subbundle of TM ⊕ T ⋆M
then the following expressions are equivalent:
• L is involutive
• NijL = 0
• JacL = 0.
Definition 8 A Dirac structure on TM⊕T ⋆M is a max-imal isotropic and involutive subbundle L < TM⊕T ⋆M .
It follows from the properties above that the Dirac struc-ture is given by NijL = 0. The advantage of using theDirac structures is its generality in contrast to structuresused in Poisson geometry, complex structures, foliated orsymplectic geometries. This Dirac structure has there-fore a great unifying power. The H- deformed Diracstructures include also generalized complex structureswhich are well defined on some manifolds without anycomplex or symplectic structures. Moreover, this kind ofgeometry became extremely important in string theory(flux compactification, mirror symmetry, branes in YMmanifolds) and related WZW models. This H- deformedDirac structures are also important for the recognition ofexotic smoothness in topological trivial case of R4. Themain idea behind this recognition is the suitable modifi-cation of the Lie product of fields on smooth manifolds.63
Firstly the modification is given by the Courant bracketon TM ⊕ T ⋆M and then by the H-deformation of it.
In differential geometry, the Lie bracket of smooth vec-tor fields on a smooth manifold M is invariant under dif-feomorphisms, and there are no other symmetries of thetangent bundle preserving the Lie bracket. More pre-cisely, let (f, F ) be a pair of diffeomorphisms f : M → Mand F : TM → TM and F is linear on each fiber, πbe the canonical projection π : TM → M . Supposethat F preserves the Lie bracket [ , ], i.e. F ([X,Y ]) =[F (X), F (Y )] for vector fields X , Y on M , and supposethe naturalness of (f, F ) i.e. π F = f F , then F hasto be equal to df .
In case of our extended ,,tangent space” TM ⊕ T ⋆M ,the Courant bracket and the inner product are diffeomor-phisms invariant. However, there exists another sym-metry extending the diffeomorphisms which is the so-called B- field transformation. Let us see how thiswork. Given a two-form B on M one can think of itas the map TM → T ⋆M by contracting B with X ,X → iXB. The transformation of TM ⊕ T ⋆M is givenby eB : X + ξ → X + ξ+ iXB with the properties (see43,Propositions 3.23, 3.24)
• The map eB is an automorphism of the Courantbracket if and only if B is closed, i.e. dB = 0,
• The eBextension of diffeomorphisms are the onlyallowed symmetries of the Courant bracket.
17
which means, that for a pair (f, F ) consisting ofa (orthogonal) automorphism of TM ⊕ T ⋆M anda transformation F preserving the Courant bracket[ , ], i.e. F ([A,B]) = [F (A), F (B)] for all sectionsA,B ∈C∞(TM → T ⋆M). Therefore F has to be a com-position of a diffeomorphism of M and a B- field trans-form. This means that the group of orthogonal Courantautomorphisms of TM ⊕ T ⋆M is the semi-direct productof Diff(M) and Ω2
closed.Given a Courant bracket on TM⊕T ⋆M we are able to
define various involutive structures by using this bracket.The most important possibility is the deformation of theCourant bracket on TM ⊕ T ⋆M by a real closed 3-formH on M . For any real 3-form H one has the twistedCourant bracket on TM ⊕ T ⋆M defined as
[X + ξ, Y + η]H = [X + ξ, Y + η] + iY iXH (30)
where [ , ] on the RHS is the non-twisted Courant bracket.This can be also restated as the splitting condition in thenon-trivial twisted Courant algebroid defined later.
This deformed bracket allows for defining various in-volutive and (maximal) isotropic structures with respectto [ , ]H , which is again an analog for the integrabilityof distributions on manifolds. These structures corre-spond to new H-twisted geometries which are differentfor the previously considered Dirac structures in case ofthe untwisted Courant bracket. In particular, the B- fieldtransform of [ , ]H is the symmetry of the bracket if andonly if dB = 0, since it yields
[eB(C), eB(D)
]H
= eB [C,D]H+dB , (31)
for all C,D ∈ C∞(TM⊕T ⋆M). Then the tangent bundleTM is not involutive with respect to [ , ]H for non-zeroH. In general a subbundle L is involutive with respect to[ , ]H if and only if e−BL is involutive for [ , ]H+dB.
In the following we will comment on the relation be-tween Hitchin structures and gerbes used in the next the-orem. Furthermore we are able to understand a way howTM ⊕ T ⋆M appears from the broader perspective of theextensions of bundles. For that purpose one defines theCourant algebroid E as an extension of real vector bun-dles given by the sequence
0 → T ⋆M →π⋆ E →π TM → 0 (32)
On E a non-degenerate symmetric bilinear form 〈 , 〉 isgiven, such that 〈π⋆ξ, a〉 = ξ(π(a)) where ξ is smoothcovector field on M and a ∈ C∞(E) . A bilinear Courantbracket [, ] on C∞(E) can be defined such that
• [a, [b, c]] − [[a, b], c] + [b, [a, c]] = 0 (Jacobi identity)
• [a, fb] = f [a, b] + (π(a)f)b (Leibniz rule)
• π(a)〈b, c〉 = 〈[a, b], c〉 + 〈b, [a, c]〉 (Invariance of bi-linear form)
• [a, a] = π⋆d〈a, a〉
The sequence (32) defines a splitting of E. Each splittingdetermines a closed 3-form H ∈ Ω3(M), given by
(iX iYH)(Z) = 〈[s(X), s(Y )], s(Z)〉 (33)
where s : TM → E is the splitting derived from thesequence. The cohomology class [H ]/2π ∈ H3(M,R) isindependent of the choice of splitting, and coincides withthe image of the Dixmier-Douady class of the gerbe inreal cohomology46. The Dixmier-Douady class classifiesthe bundle gerbes as done in Section V.
Theorem 8 Given a radial family of small exotic R4 andradius r, the Godbillon-Vey invariant is an element ofGV ∈ H3(S3,R) ≈ R and GV = r2. The 3-sphere S3
used to construct the boundary ∂A of the Akbulut cork,induces an isomorphism H3(S3,R) ≃ H3(∂A,R). Onehas the following correspondences:
• A. The continuous family of distinct small exoticsmooth structures on R4 corresponds to the H-deformed classes of generalized Hitchin’s geometrieson S3, where [H ] ∈ H3(S3,R), as well of the bound-ary of the Akbulut corkΣ = ∂A, so that [HΣ] ∈H3(Σ,R) is the deformation of the Hitchin geom-etry on Σ. Every deformation [H ] ∈ H3(S3,R)can be realized by a noncobordant codimension-1Thurston-type foliation of the 3-sphere S3 and alsoby one exotic R4 of the radial family. Furthermorethis foliation defines also a Dirac structure.
• B. For integral classes [H ] ∈ H3(S3,Z) these de-formations are geometrically described by S1gerbes.(see also theorem 7)
Proof: Theorem 8: We start to show the correspon-dence A. The class H ∈ H3(S3,R) to deform the Courantbracket on TS3 ⊕ T ⋆S3 will be used to relate it with thesmall exotic R4 by theorem 5. Because of the isomor-phism S3 = SU(2), the tangent bundle TS3 is trivial, i.e.TS3 = S3 × R3 and there are three global vector fields(as sections) generating (locally) every other vector fieldby linear combination. The triviality of H2(S3,R) = 0implies that Ω2
closed are given by all exact 2-forms as im-age of all 1-forms Ω2
closed = dΩ1. Therefore the auto-morphisms are mainly generated by the diffeomorphismDiff(S3). Now we consider the 1-form ω defining theThurston-type foliation on S3 with the dual X = ω∗ , avector field. This vector field generates a 1-dimensionalsubbundle ⊂ TS3 so that ⊕Ann() ⊂ TS3⊕T ∗S3
(Ann() is the annihilator) is a maximal isotropic sub-bundle, hence defines a Dirac structure (see43 Example3.32). Now we choose the closed 3-form H = ΓF withthe Godbillon-Vey class ΓF as deformation of the Hitchinstructure. We know from theorem 5 that the radial fam-ily of small exotic R4’s is determined by the noncobor-dant codimension-1 foliations of Σ and by theorem 4 it
18
is also determined by noncobordant codimension-1 foli-ations of S3 as well. The surjectivity of the homomor-phism (9) implies that every 3-form can be realized by theGodbillon-Vey class of at least one foliation of Thurstontype (up to foliated cobordisms). The discussion canalso extended for the 3-manifold Σ showing the full cor-respondence A. Let us mention the possibility to takegeneralized complex structures on S3 × R by using theisomorphism H3(S3 × R,R) ≃ H3(S3,R) by consideringa specific embedding of the collar into the neighborhoodN(A) ⊂ R4 of the Akbulut cork A.
To show the correspondence B, we will use the com-ments about the Courant algebroid above. The condition(21) in Sec. V for the S1-gerbe with connection has theinterpretation that dAαβ is a cocycle. Using this inter-pretation, one can locally paste together (TM ⊕ T ⋆M)α
and (TM ⊕ T ⋆M)β by the automorphism
(1 0
dAαβ 1
).
The action of dAαβ on TM is defined by X → iXdAαβ .The second condition for the connection data of a gerbe(see (22) in Sec. V) defines a splitting of the Courantalgebroid like in (33). Then we can make use of theproposition 3.47 in43:If [H/2π] ∈ H3(M,Z) then the twisted Courant bracket[, ]H on TM⊕T ⋆M can be obtained from a S1 gerbe withconnection.If [H/2π] is integral, then because of dB = 0 and (31)the trivializations fαβ of a flat gerbe with connection aresymmetries of [, ]H (B-field transforms). The differenceof two possible trivializations is a line bundle with con-nection see Sec. V. These line bundles have the role ofgauge transformations (integral B-fields)43 establishingthe correspondence B.
A general result for the family of small smooth R4’scan be restated asThe change of exotic smooth structure on R4 results inthe change of a generalized Dirac structures on S3 lyingat the boundary of the Akbulut cork in R4,and composing with the conjecture from Sec. IV G wehave a geometrical interpretationThe change of smoothness on S4 corresponds to the de-formation of a generalized Dirac structure by S1- gerbeson S3 lying at the boundary of the Akbulut cork for S4.It is also possible to conjecture a connection to 2-gerbes:For non-integral [H ] the geometrical description requiresS1 (fully abelian) categorical 2-gerbes.For that purpose, we note that non-integral third coho-mologies can be geometrically interpreted by gerbes. Butthen one needs a 2-categorical extension of these gerbes,namely fully abelian 2-gerbes5. In this work we omittedthe usage of a categorical language and kept the pre-sentation down to earth from that point of view. Theconnection of exotic smoothness in dimension four and(higher) category theory constructions will be presentedin a separate work.
VII. AN APPLICATION: CHARGE QUANTIZATIONWITHOUT MAGNETIC MONOPOLES
The connection between the small exotic R4 con-structed from the radial family and the codimension-1foliations of S3 with the Godbillon-Vey class representedby 3-rd cohomologiesH3(S3,R) enables one to realize ge-ometrical effects of these exotica in R4 via abelian gerbeson S3 and twisted Courant brackets. The direct rela-tion between the integral classes in H3(S3,Z) and exoticR4was discussed in subsection IV F. In this section wewill discuss this relation as a kind of localization principlecausing strong physical consequences. Namely, the condi-tion for magnetic monopoles in spacetime is expressed interms of abelian gerbes which gives non-vanishing thirdintegral cohomologies H3(S3,Z) (26, Chapter 7). As wesaw in Sec. IV the change of smoothness on R4 betweenthe members of the radial family for integer radii, corre-sponds to the change between the corresponding integralclasses H3(S3,Z). Here we will state the stronger local-ization principle:Effects of exotic smooth R4 in a region of the 4-
spacetime M4 can be localized to S3 embedded in M4,i.e. the appearance of this exotic smoothness is equivalentto the appearance of the nontrivial, geometrically realizedclass in H3(S3,Z). This class is stable with respect tocontinuous deformations.
Here we will discuss the physical consequence of theexoticness of an open 4-region in 4-spacetime:The quantization condition for electric charge in space-
time can be seen as a consequence of certain non-standard4-smoothness appearing in spacetime.In the following we will present the details to understandthis consequence.
A. Dirac’s magnetic monopoles and S1 - gerbes
When the magnetic field B is defined over the wholeeuclidean space R3, there exists a globally defined vec-tor potential, and any two vector potentials differ by thegradient of some function. In terms of the connection onthe line bundle L, one can trivialize the bundle, and theconnection ∇ on L is given by
∇ = d+ ie
~cA (34)
Dirac considered a magnetic field B defined on R3 \ 0which has a singularity at the origin. This singularitycorresponds to the existence of a magnetic monopole lo-calized at the origin. The magnetic monopole has thestrength µ
µ =1
4π
∫ ∫
S2
−→B × dσ (35)
19
which is the flux of−→B through the 2-sphere S2 up to a
constant. Because of div(−→B ) = 0, the integral does not
depend on the choice of the Σ centered at the origin.Equivalently µ can be expressed in terms of the curva-
ture 2-form R, of a connection on L, as
µ = −i c~4πe
∫
S2
R (36)
Now, the integral∫S2 R has values which are integer mul-
tiplicities of 2πi on a complex line bundle L. Thus, thefollowing quantization condition for the strength of amagnetic monopole, follows
µ =c~
2e· n, n ∈ Z (37)
The whole discussion based on the fact that the coho-mology class of R
2πi is integral and the magnetic field−→B
is proportional to the curvature of some line bundle withconnection on R3 \ 0. We see, that (37) is the same as2c~µ · e = n and this means that the electrical charge has
to be quantized.The cohomologies involved here are H2(R3 \0) . We
extend, following26, Chap. 7, the forms and cohomologiesover the whole 3-space, including the origin. To this endlet us consider generalized 3-forms on R3 which are sup-ported at the origin, i.e. the relative cohomology groupH3(R3,R3 \ 0) =: H3
0 (R3) . Given the exact sequence
H2(R3) = 0 → H2(R3 \ 0) → H30 (R3) → H3(R3) = 0
(38)we have the isomorphism
H2(R3 \ 0) = H30 (R3) (39)
We saw in (37) that a topological analog of themonopole is the element of the 2-nd cohomology H2(R3 \0 ,Z). Thus, the extension of the description of amonopole, located at the origin, over entire R3, givesthe topological analog of the monopole as an element ofH3
0 (R3). A monopole is moving now inside the 3-space,and from the canonical isomorphisms
H30 (R3) ≃ H3
0 (R3 ∪ ∞) ≃ H3(S3) (40)
the topological counterpart of a monopole is the elementof H3(S3,Z). Conversely, the topological realization ofsome element in H3(S3,Z) is given by some line bundlewith connection on R3 \ 0, which is equivalent to theexistence of a Dirac monopole in spacetime and conse-quently the electric charge is quantized.
The whole discussion can be extended to the relativis-tic theory in R4 as well (see the introduction of21). Thenwe consider the Coulomb potential of a point particle of
charge q in 0 ∈ R4 as the connection one-form of a linebundle
A = −q · 1
r· dt
over R4 \ Rt with r2 = x2 + y2 + z2 6= 0. The curvatureis given by the exact 2-form
F = dA
fulfilling the first Maxwell equation dF = 0. Now weconsider the 2-form ∗F
∗F =q
4π· x dy ∧ dz − y dx ∧ dz + z dx ∧ dy
r3
as element of H2(R4 \ Rt) fulfilling the second Maxwellequation d ∗ F = 0. A simple argument showed the iso-morphism
H∗(R4 \ Rt) ∼= H∗(R3 \ 0)
and the integral along an embedded S2 surrounding thepoint 0 gives
∫
S2
∗F = q
the charge q of the particle. Together with the isomor-phism (39) we obtain the relation between a relativisticparticle of charge q in R4 and elements of H3(S3).
The elements of H3(S3,Z) have well-defined topologi-cal realizations like the elements of H2(R3 \ 0 ,Z) cor-responding to line bundles with connection. Namely,h ∈ H3(S3,Z) corresponds to the S1- gerbe Gh on S3.However, the elements of H3(S3,Z) have yet another re-alizations in spacetime.
B. S1-gerbes on S3 and exotic smooth R4’s
In Sec. IV it was shown that third real deRham coho-mology classes of S3 correspond to the isomorphic classesof codimension-1 foliations of S3 (see Theorem 5). Forintegral 3-rd cohomologies of S3 we have the correspon-dence between S1- gerbes on S3 and some exotic smoothR4’s. The correspondence means, in particular, thatsome exotic smooth small structures on some open regionin R4 serves as the spacetime realization of the integral3-rd cohomology class of S3. In other words, a non-trivial3-rd cohomology class of S3 is realized in spacetime byexotic smooth 4-structure in some region. On the otherhand, a class in H3(S3,Z) (canonically isomorphic toH2(S2,Z) ≃ H2(R3\0 ,Z)) is realized geometrically asa line bundle with connection on R3\0. From the pointof physics in 4-spacetime, the existence of this line bundlecould be equivalent to the action of a monopole existingsomewhere in spacetime. However, if a monopole exists,electric charge is quantized. We have to make some addi-tional assumptions to be sure that exoticness of a region
20
of spacetime can give the same effect as the existenceof a monopole. Namely, we suppose that the magneticfield propagates over a region with an exotic smoothnessstructure in 4-spacetime with Minkowskie metric suchthat
• the exotic smooth R4h corresponds to the class [h] ∈
H3(S3,Z) and
• the strength of the magnetic field is proportional tothe curvature of the line bundle on R3\0 whichcorresponds to this [h] ∈ H2(R3 \ 0).
Then this exotic R4h acts as a source for the magnetic
field in R4, i.e. the magnetic charge of the monopole andthe electric charge are quantized, see (37). Given a class[h] ∈ H2(R3\0) we always find a monopole solution ful-filling these requirements. By (37), a monopole gives thequantization of the electric charge and is the source forthe magnetic field. As remarked in subsection IV F, theThurston-type foliation leading to integer Godbillon-Veynumbers is stable with respect to deformations. There-fore, the quantization is stable with respect to continuousdeformations as well.
As conjectured by Brans23, some large exotic smoothR4’s can act as the external sources of gravitational fieldin spacetime. This Brans conjecture states that exoticsmooth structures on 4-manifolds (compact and non-compact) can serve as external sources for gravitationalfield22. The Brans conjecture was proved by Asselmeyer7
in the compact case, and by S ladkowski76 in the non-compact case.
In that way, we yielded an essential extension of theBrans conjecture for magnetic fields:Some small, exotic smooth structures on R4 can act as
sources of magnetic field, i.e. monopoles, in spacetime.Electric charge in spacetime has to be quantized, providedsome region has this small exotic smoothness.
The connection of small exotic R4 with magnetism andspins of particles is especially important and opens a widerange of physical applications. We will discuss this ideain a forthcoming paper.
ACKNOWLEDGMENT
We thank L. Taylor for the discussion of an error inour argumentation. T.A. wants to thank C.H. Brans andH. Rose for numerous discussions over the years aboutthe relation of exotic smoothness to physics. J.K. bene-fited much from the explanations given to him by RobertGompf regarding 4-smoothness several years ago, anddiscussions with Jan S ladkowski.
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