Constructing a quantum field theory from spacetime
The paper shows deep connections between exotic smoothings of a small (the spacetime), the leaf space of codimension-1 foliations (related to noncommutative algebras) and quantization. At first we relate a small exotic to codimension-1 foliations of the 3-sphere unique up to foliated cobordisms and characterized by the real-valued Godbillon-Vey invariant. Special care is taken for the integer case which is related to flat bundles. Then we discuss the leaf space of the foliation using noncommutative geometry. This leaf space contains the hyperfinite factor of Araki and Woods important for quantum field theory (QFT) and the factor. Using Tomitas modular theory, one obtains a relation to a factor algebra given by the horocycle foliation of the unit tangent bundle of a surface of genus . The relation to the exotic is used to construct the (classical) observable algebra as Poisson algebra of functions over the character variety of representations of the fundamental group into the . The Turaev-Drinfeld quantization (as deformation quantization) of this Poisson algebra is a (complex) skein algebra which is isomorphic to the hyperfinite factor algebra determining the factor algebra of the horocycle foliation. Therefore our geometrically motivated hyperfinite factor algebra comes from the quantization of a Poisson algebra. Finally we discuss the states and operators to be knots and knot concordances, respectively.
- 1 Introduction
- 2 Preliminaries: Foliations and Operator algebras
3 Exotic and codimension-one foliations
- 3.1 Smoothness on manifolds
- 3.2 Small exotic ’s and Akbulut corks
- 3.3 Exotic and Casson handles
- 3.4 The design of a Casson handle and its foliation
- 3.5 Capped gropes and its design
- 3.6 The radial family of uncountable-many small exotic
- 3.7 Exotic and codimension-1 foliations
- 3.8 Integer Godbillon-Vey invariants and flat bundles
- 3.9 From exotic smoothness to operator algebras
- 4 The connection between exotic smoothness and quantization
- 5 Knots as states
- 6 Discussion
- A Proof of Theorem 3.3
The construction of quantum theories from classical theories, known as quantization, has a long and difficult history. It starts with the discovery of quantum mechanics in 1925 and the formalization of the quantization procedure by Dirac and von Neumann. The construction of a quantum theory from a given classical one is highly non-trivial and non-unique. But except for few examples, it is the only way which will be gone today. From a physical point of view, the world surround us is the result of an underlying quantum theory of its constituent parts. So, one would expect that we must understand the transition from the quantum to the classical world. But we had developed and tested successfully the classical theories like mechanics or electrodynamics. Therefore one tried to construct the quantum versions out of classical theories. In this paper we will go the other way to obtain a quantum field theory by geometrical methods and to show its equivalence to a quantization of a classical Poisson algebra.
The main technical tool will be the noncommutative geometry developed by Connes Con:85 (). Then intractable space like the leaf space of a foliation can be described by noncommutative algebras. From the physical point of view, we have now an interpretation of noncommutative algebras (used in quantum theory) in a geometrical context. So, we need only an idea for the suitable geometric structure. For that purpose one formally considers the path integral over spacetime geometries. In the evaluation of this integral, one has to include the possibility of different smoothness structures for spacetime Pfeiffer2004 (); Ass2010 (). Brans BraRan:93 (); Bra:94a (); Bra:94b () was the first who considered exotic smoothness also on open smooth 4-manifolds as a possibility for space-time. He conjectured that exotic smoothness induces an additional gravitational field (Brans conjecture). The conjecture was established by Asselmeyer Ass:96 () in the compact case and by Sładkowski Sladkowski2001 () in the non-compact case. Sładkowski Sla:96 (); Sla:96b (); Sla:96c () discussed the influence of differential structures on the algebra of functions over the manifold with methods known as non-commutative geometry. Especially in Sla:96b (); Sla:96c () he stated a remarkable connection between the spectra of differential operators and differential structures. But there is a big problem which prevents progress in the understanding of exotic smoothness especially for the : there is no known explicit coordinate representation. As the result no exotic smooth function on any such is known even though there exist families of infinite continuum many different non diffeomorphic smooth . This is also a strong limitation for the applicability to physics of non-standard open 4-smoothness. Bizaca Biz:94 () was able to construct an infinite coordinate patch by using Casson handles. But it still seems hopeless to extract physical information from that approach.
This situation is not satisfactory but we found a possible solution. The solution is a careful analysis of the small exotic by using foliation theory (see next section) to derive a relation between exotic smoothness and codimension-1 foliations in section 3 (see Theorem 3.3). By using noncommutative geometry, this approach is able to produce a von Neumann algebra via the leaf space of the foliation which can be interpreted as the observable algebra of some QFT (see Haag:96 ()). Fortunately, our approach to exotic smoothness is strongly connected with a codimension-1 foliation of type , i.e. the leaf space is a factor von Neumann algebra. Especially this algebra is the preferred algebra in the local algebra approach to QFT Haag:96 (); Borchers2000 (). Recently, this factor case was also discussed in connection with quantum gravity (via the spectral triple of Connes) BertozziniConti2010 ().
In the next two sections we will give an overview about foliation theory, its operator-theoretical description and the relation to exotic smoothness. Both sections are rather technical with a strong overlap to our previous paper AsselmeyerKrol2009 (). In section 4 we turn to the quantization procedure as related to nonstandard smoothings of . Based on the dictionary between operator algebra and foliations one has the corresponding relation of small exotic ’s and operator algebras. This is a noncommutative algebra which can be seen as the algebra of quantum observables of some theory.
First, in subsection 2.5 we recognized the algebra as the hyperfinite factor von Neumann algebra. From Tomita-Takesaki theory it follows that any factor algebra decomposes as a crossed product into where is a factor . Via Connes procedure one can relate the factor foliation to a factor foliation. Then we obtain a foliation of the horocycle flow on the unit tangent bundle over some genus surface which determines the factor . This foliation is in fact determined by the horocycles which are closed circles.
Next we are looking for a classical algebraic structure which would give the above mentioned noncommutative algebra of observables as a result of quantization. The classical structure is recovered by the idempotent of the algebra and has the structure of a Poisson algebra. The idempotents were already constructed in subsection 2.4 as closed curves in the leaf of the foliation of . As noted by Turaev Turaev1991 (), closed curves in a surface induce a Poisson algebra: Given a surface let be the space of flat connections of bundles on ; this space carries a Poisson structure as is shown in subsection 4.2. The complex functions on can be considered as the algebra of classical observables forming the Poisson algebra .
Next in the subsection 4.3 we find a quantization procedure of the above Poisson algebra which is the Drinfeld-Turaev deformation quantization. It is shown that the result of this quantization is the skein algebra for the deformation parameter ( corresponds to the commutative Poisson structure on ).
This skein algebra is directly related to the factor von Neumann algebra derived from the foliation of . In fact the skein algebra is constructed in subsection 4.4 as the factor algebra Morita equivalent to the factor which in turn determines the factor of the foliation.
Finally in section 5 we discuss the states of the algebra and the operators between states. Here, we present only the ideas: the states are knots represented by holonomies along a flat connection. Then an operator between two states is a knot concordance (a kind of knot cobordism). The whole approach is similar to the holonomy flux algebra of Loop quantum gravity (see Thiemann2006 ()). We will discuss this interesting relation in our forthcoming work.
2 Preliminaries: Foliations and Operator algebras
In this section we will consider a foliation of a manifold , i.e. an integrable subbundle of the tangent bundle . The leaves of the foliation are the maximal connected submanifolds with . We denote with the set of leaves or the leaf space. Now one can associate to the leaf space a algebra by using the smooth holonomy groupoid of the foliation (see Connes Connes1984 ()). For a codimension-1 foliation of a 3-manifold there is the Godbillon-Vey invariant GodVey:71 () as element of . As example we consider the construction of a codimension-1 foliation of the 3-sphere by Thurston Thu:72 () which will be used extensively in the paper. This foliation has a non-trivial Godbillon-Vey invariant where every element of is represented by a cobordism class of foliations. Hurder and Katok HurKat:84 () showed that the algebra of a foliation with non-trivial Godbillon-Vey invariant contains a factor subalgebra. In the following we will construct this algebra and discuss the factor case.
2.1 Definition of Foliations and foliated cobordisms
A codimension foliation111In general, the differentiability of a foliation is very important. Here we consider the smooth case only. of an -manifold (see the nice overview article Law:74 ()) is a geometric structure which is formally defined by an atlas , with , such that the transition functions have the form
Intuitively, a foliation is a pattern of -dimensional stripes - i.e., submanifolds - on , called the leaves of the foliation, which are locally well-behaved. The tangent space to the leaves of a foliation forms a vector bundle over , denoted . The complementary bundle is the normal bundle of . Such foliations are called regular in contrast to singular foliations or Haefliger structures. For the important case of a codimension-1 foliation we need an overall non-vanishing vector field or its dual, an one-form . This one-form defines a foliation iff it is integrable, i.e.
and the leaves are the solutions of the equation
Now we will discuss an important equivalence relation between foliations, cobordant foliations. Let and be two closed, oriented -manifolds with codimension- foliations. Then these foliated manifolds are said to be foliated cobordant if there is a compact, oriented -manifold with boundary and with a codimension- foliation transverse to the boundary and inducing the given foliation there. The resulting foliated cobordism classes form a group under disjoint union.
2.2 Non-cobordant foliations of detected by the Godbillon-Vey class
In Thu:72 (), Thurston constructed a foliation of the 3-sphere depending on a polygon in the hyperbolic plane so that two foliations are non-cobordant if the corresponding polygons have different areas. For later usage, we will present this construction now (see also the book Tamura1992 () chapter VIII for the details).
Consider the hyperbolic plane and its unit tangent bundle , i.e the tangent bundle where every vector in the fiber has norm . Thus the bundle is a -bundle over . There is a foliation of invariant under the isometries of which is induced by bundle structure and by a family of parallel geodesics on . The foliation is transverse to the fibers of . Let be any convex polygon in . We will construct a foliation of the three-sphere depending on . Let the sides of be labeled and let the angles have magnitudes . Let be the closed region bounded by , where is the reflection of through . Let , be minus an open -disk about each vertex. If is the projection of the bundle , then is a solid torus (with edges) with foliation induced from . For each , there is an unique orientation-preserving isometry of , denoted , which matches point-for-point with its reflected image . We glue the cylinder to the cylinder by the differential for each , to obtain a manifold , and a (glued) foliation , induced from . To get a complete , we have to glue-in solid tori for the Now we choose a linear foliation of the solid torus with slope (Reeb foliation). Finally we obtain a smooth codimension-1 foliation of the 3-sphere depending on the polygon .
Now we consider two codimension-1 foliations depending on the convex polygons and in . As mentioned above, these foliations are defined by two one-forms and with and . Now we define the one-forms as the solution of the equation
and consider the closed 3-form
associated to the foliation . As discovered by Godbillon and Vey GodVey:71 (), depends only on the foliation and not on the realization via . Thus , the Godbillon-Vey class, is an invariant of the foliation. Let and be two cobordant foliations then . In case of the polygon-dependent foliations , Thurston Thu:72 () obtains
is cobordant to
and are non-cobordant
We note that . The Godbillon-Vey class is an element of the deRham cohomology which will be used later to construct a relation to gerbes. Furthermore we remark that the classification is not complete. Thurston constructed only a surjective homomorphism from the group of cobordism classes of foliation of into the real numbers . We remark the close connection between the Godbillon-Vey class (1) and the Chern-Simons form if can be interpreted as connection of a suitable line bundle.
2.3 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 foliations of a homology 3-sphere . Because of the diffeomorphism , we can relate a foliation on to a foliation on . By using the surgery along a knot or link, we are able to construct the codimension-one foliation for every compact 3-manifold.
Given a compact 3-manifold without boundary. Every codimension-one foliation of the 3-sphere (constructed above) induces a codimension-one foliation on . For every cobordism class as element of the deRham cohomology , there exists an element of with a cobordism class .
The proof can be found in AsselmeyerKrol2009 ().
2.4 The smooth holonomy groupoid and its algebra
Let be a foliated manifold. Now we shall construct a von Neumann algebra canonically associated to and depending only on the Lebesgue measure class on the space of leaves of the foliation. In the following we will identify the leaf space with this von Neumann algebra. The classical point of view, , will only give the center of . According to Connes Connes94 (), we assign to each leaf the canonical Hilbert space of square-integrable half-densities . This assignment, i.e. a measurable map, is called a random operator forming a von Neumann . The explicit construction of this algebra can be found in Connes1984 (). Here we remark that is also a noncommutative Banach algebra which is used above. Alternatively we can construct as the compact endomorphisms of modules over the algebra of the foliation also known as holonomy algebra. From the point of view of K theory, both algebras and are Morita-equivalent to each other leading to the same groups. In the following we will construct the algebra by using the holonomy groupoid of the foliation.
Given a leaf of and two points of this leaf, any simple path from to on the leaf uniquely determines a germ of a diffeomorphism from a transverse neighborhood of to a transverse neighborhood of . The germ of diffeomorphism thus obtained only depends upon the homotopy class of in the fundamental groupoid of the leaf , and is called the holonomy of the path . The holonomy groupoid of a leaf is the quotient of its fundamental groupoid by the equivalence relation which identifies two paths and from to (both in ) iff . The holonomy covering of a leaf is the covering of associated to the normal subgroup of its fundamental group given by paths with trivial holonomy. The holonomy groupoid of the foliation is the union of the holonomy groupoids of its leaves.
Recall a groupoid is a category where every morphism is invertible. Let be a set of objects and the set of morphisms of , then the structure maps of reads as:
where is the composition of the composable two morphisms (target of the first is the source of the second), is the inversion of an arrow, the source and target maps respectively, assigns the identity to every object. We assume that are smooth manifolds and all structure maps are smooth too. We require that the maps are submersions, thus is a manifold as well. These groupoids are called smooth groupoids.
Given an element of , we denote by the origin of the path and its endpoint with the range and source maps . An element of is thus given by two points and of together with an equivalence class of smooth paths: the , with and , tangent to the bundle (i.e. with , ) identifying and as equivalent iff the holonomy of the path at the point is the identity. The graph has an obvious composition law. For , the composition makes sense if . The groupoid is by construction a (not necessarily Hausdorff) manifold of dimension . We state that is a smooth groupoid, the smooth holonomy groupoid.
Then the algebra of the foliation is the algebra of the smooth holonomy groupoid . For completeness we will present the explicit construction (see Connes94 () sec. II.8). The basic elements of ) are smooth half-densities with compact supports on , , where for is the one-dimensional complex vector space , where , and is the one-dimensional complex vector space of maps from the exterior power ,, to such that
For , the convolution product is given by the equality
Then we define via a operation making into a algebra. For each leaf of one has a natural representation of on the space of the holonomy covering of . Fixing a base point , one identifies with and defines the representation
The completion of with respect to the norm
makes it into a algebra . Among all elements of the algebra, there are distinguished elements, idempotent operators or projectors having a geometric interpretation in the foliation. For later use, we will construct them explicitly (we follow Connes94 () sec. closely). Let be a compact submanifold which is everywhere transverse to the foliation (thus ). A small tubular neighborhood of in defines an induced foliation of over with fibers . The corresponding algebra is isomorphic to with the algebra of compact operators. In particular it contains an idempotent , , where is a minimal projection in . The inclusion induces an idempotent in . Now we consider the range map of the smooth holonomy groupoid defining via a submanifold. Let be a section (with compact support) of the bundle of half-density over so that the support of is in the diagonal in and
Then the equality
defines an idempotent . Thus, such an idempotent is given by a closed curve in transversal to the foliation.
2.5 Some information about the factor case
In our case of codimension-1 foliations of the 3-sphere with nontrivial Godbillon-Vey invariant we have the result of Hurder and Katok HurKat:84 (). Then the corresponding von Neumann algebra contains a factor algebra. At first we will give an overview about the factor .
Remember a von Neumann algebra is an involutive subalgebra of the algebra of operators on a Hilbert space that has the property of being the commutant of its commutant: . This property is equivalent to saying that is an involutive algebra of operators that is closed under weak limits. A von Neumann algebra is said to be hyperfinite if it is generated by an increasing sequence of finite-dimensional subalgebras. Furthermore we call a factor if its center is equal to . It is a deep result of Murray and von Neumann that every factor can be decomposed into 3 types of factors . The factor case divides into the two classes and with the hyperfinite factors the complex square matrices and the algebra of all operators on an infinite-dimensional Hilbert space . The hyperfinite factors are given by , the Clifford algebra of an infinite-dimensional Euclidean space , and . The case remained mysterious for a long time. Now we know that there are three cases parametrized by a real number : the Krieger factor induced by an ergodic flow , the Powers factor for and the Araki-Woods factor for all with . We remark that all factor cases are induced by infinite tensor products of the other factors. One example of such an infinite tensor space is the Fock space in quantum field theory.
But now we are interested in an explicit construction of a factor von Neumann algebra of a foliation. The interesting example of this situation is given by the Anosov foliation of the unit sphere bundle of a compact Riemann surface of genus endowed with its Riemannian metric of constant curvature . In general the manifold is the quotient of the semi-simple Lie group , the isometry group of the hyperbolic plane , by the discrete cocompact subgroup , and the foliation of is given by the orbits of the action by left multiplication on of the subgroup of upper triangular matrices of the form
The von Neumann algebra of this foliation is the (unique) hyperfinite factor of type . In the subsection 2.2 we describe the construction of the codimension-1 foliation on the 3-sphere . The main ingredient of this construction is the convex polygon in the hyperbolic plane having curvature . The Reeb components of this foliation of are represented by a factor algebra and thus do not contribute to the Godbillon-Vey class. Putting all things together we will get
The codimension-1 foliation of the 3-sphere with non-trivial Godbillon-Vey invariant is also associated to a von Neumann algebra induced by the foliation which contains a factor algebra, the hyperfinite factor .
This theorem follows mostly from the work Hurder and Katok HurKat:84 (). The codimension-1 foliation of the 3-sphere was constructed in subsection 2.2. It admits a non-trivial Godbillon-Vey invariant related to the volume of the polygon in . The whole construction do not depend on the number of vertices of but on the volume only. Thus without loss of generality, we can choose the even number for of vertices for . As model of the hyperbolic plane we choose the usual upper half-plane model where the group (the real Möbius transformations) and the hyperbolic group (the group of all orientation-preserving isometries of ) act via fractional linear transformations. Then the polygon is a fundamental polygon representing a Riemann surface of genus . Via the procedure above, we can construct a foliation on with . This foliation is also induced from the foliation of (as well as the foliation of the ) via the left action above. The difference between the foliation on and on is given by the different usage of the polygon . Thus the von Neumann algebra of the codimension-1 foliation of the 3-sphere contains a factor algebra in agreement with the results in HurKat:84 (). In the notation above we have the unit tangent bundle of the polygon equipped with an Anosov foliation (see also Tamura1992 ()). The group acts as isometry on where the modular group acts as discrete subgroup leaving the polygon (seen as fundamental domain) invariant. The upper triangular matrices above are elements of and act by linear fractional transformation inducing a shift. The orbits of this action have therefore constant velocity (the horocycle flow) and we are done.
We showed that this factor algebra is the hyperfinite factor . Now one may ask, what is the physical meaning of the factor ? Because of the Tomita-Takesaki-theory, factor algebras are deeply connected to the characterization of equilibrium temperature states of quantum states in statistical mechanics and field theory also known as Kubo-Martin-Schwinger (KMS) condition. Furthermore in the quantum field theory with local observables (see Borchers Borchers2000 () for an overview) one obtains close connections to Tomita-Takesaki-theory. For instance one was able to show that on the vacuum Hilbert space with one vacuum vector the algebra of local observables is a factor algebra. As shown by Thiemann et. al. Thiemann2006 () on a class of diffeomorphism invariant theories there exists an unique vacuum vector. Thus the observables algebra must be of this type.
3 Exotic and codimension-one foliations
Einsteins insight that gravity is the manifestation of geometry leads to a new view on the structure of spacetime. From the mathematical point of view, spacetime is a smooth 4-manifold endowed with a (smooth) metric as basic variable for general relativity. Later on, the existence question for Lorentz structure and causality problems (see Hawking and Ellis HawEll:94 ()) gave further restrictions on the 4-manifold: causality implies non-compactness, Lorentz structure needs a codimension-1 foliation. Usually, one starts with a globally foliated, non-compact 4-manifold fulfilling all restrictions where is a smooth 3-manifold representing the spatial part. But other non-compact 4-manifolds are also possible, i.e. it is enough to assume a non-compact, smooth 4-manifold endowed with a codimension-1 foliation. All these restrictions on the representation of spacetime by the manifold concept are clearly motivated by physical questions. Among the properties there is one distinguished element: the smoothness. Usually one assumes a smooth, unique atlas of charts (i.e. a smooth or differential structure) covering the manifold where the smoothness is induced by the unique smooth structure on . But that is not the full story. Even in dimension 4, there are an infinity of possible other smoothness structures (i.e. a smooth atlas) non-diffeomorphic to each other. For a deeper insight we refer to the book Asselmeyer2007 ().
3.1 Smoothness on manifolds
If two manifolds are homeomorphic but non-diffeomorphic, they are exotic to each other. The smoothness structure is called an exotic smoothness structure.
The implications for physics are tremendous because we rely on the smooth calculus to formulate field theories. Thus different smoothness structures have to represent different physical situations leading to different measurable results. But it should be stressed that exotic smoothness is not exotic physics. Exotic smoothness is a mathematical structure which should be further explored to understand its physical relevance.
Usually one starts with a topological manifold and introduces structures on them. Then one has the following ladder of possible structures:
We do not want to discuss the first transition, i.e. the existence of a triangulation on a topological manifold. But we remark that the existence of a PL structure implies uniquely a smoothness structure in all dimensions smaller than 7 KirSie:77 (). The following basic facts should the reader keep in mind for any dimensional manifold :
The maximal differentiable atlas of is the smoothness structure.
Every manifold can be embedded in with . A smooth embedding induces the standard smooth structure on . All other possible smoothness structures are called exotic smoothness structures.
The existence of a smoothness structure is necessary to introduce Riemannian or Lorentz structures on , but the smoothness structure don’t further restrict the Lorentz structure.
3.2 Small exotic ’s and Akbulut corks
Now we consider two homeomorphic, smooth, but non-diffeomorphic 4-manifolds and . As expressed above, a comparison of both smoothness structures is given by a h-cobordism between and ( are homeomorphic). Let the 4-manifolds additionally be compact, closed and simple-connected, then we have the structure theorem222A diffeomorphism will be described by the symbol in the following. of h-cobordisms CuFrHsSt:97 ():
Let be a h-cobordisms between , then there are contractable submanifolds and a h subcobordism with , so that the remaining h-cobordism trivializes inducing a diffeomorphism between and .
In short it means that the smoothness structure of is determined by the contractable manifold – its Akbulut cork – and by the embedding of into . As shown by FreedmanFre:82 (), the Akbulut cork has a homology 3-sphere333A homology 3-sphere is a 3-manifold with the same homology as the 3-sphere . as boundary. The embedding of the cork can be derived now from the structure of the h-subcobordism between and . For that purpose we cut out from and out from . Then we glue in both submanifolds via the maps and . Both maps are involutions, i.e. . One of these maps (say ) can be chosen to be trivial (say ). Thus the involution determines the smoothness structure. Especially the topology of the Akbulut cork and its boundary is given by the topology of . For instance, the Akbulut cork of the blow-uped 4-dimensional K3 surface is the so-called Mazur manifold AkbKir:79 (); Akb:91 () with the Brieskorn-Sphere as boundary. Akbulut and its coworkers Akbulut08 (); Akbulut09 () discuss many examples of Akbulut corks and the dependence of the smoothness structure on the cork.
For the following we need a short account of the proof of the h-cobordism structure theorem. The interior of every h-cobordism can be divided into pieces, called handle Mil:65 (). A -handle is the manifold which will be glued along the boundary . The pairs of and handles in a h-cobordism between the two homeomorphic 4-manifolds and can be killed by a general procedure (Mil:65 (), §8). Thus only the pairs of handles are left. Exactly these pairs are the difference between the smooth h-cobordism and the topological h-cobordism. To eliminate the handles one has to embed a disk without self-intersections into (Whitney trick). But that is mostly impossible in 4-dimensional manifolds. Therefore Casson Cas:73 () constructed by an infinite, recursive process a special handle – the Casson-handle – containing the required disk without self-intersections. Freedman was able to show topologically the existence of this disk and he constructs a homeomorphism between every Casson handle and the open 2-handle Fre:82 (). But is in general non-diffeomorphic to as shown later by Gompf Gom:84 (); Gom:89 ().
Now we consider the smooth h-cobordism together with a neighborhood of handles. It is enough to assume a pair of handles with two self-intersections (of opposite orientation) between the 2- and 3-Spheres at the boundary of the handle. Thus one can construct an Akbulut cork in out of this data CuFrHsSt:97 (). The pair of handles can be eliminated topologically by the embedding of a Casson handle. Then as shown by Bizaca and Gompf BizGom:96 () the neighborhood of the handle pair as well the neighborhood of the embedded Akbulut cork consists of the cork and the Casson handle . Especially the open neighborhood of the Akbulut cork is an exotic . The situation was analyzed in GomSti:1999 ():
Let be a non-trivial (smooth) h-cobordism between and (i.e. is not diffeomorphic to ). Then there is an open sub-h-cobordism that is homeomorphic to and contains a compact contractable sub-h-cobordism (the cobordism between the Akbulut corks, see above), such that both and are trivial cobordisms outside of , i.e. one has the diffeomorphisms
(the latter can be chosen to be the restriction of the former). Furthermore the open sets and are homeomorphic to which are exotic if is non-trivial.
Then one gets an exotic which smoothly embeds automatically in the 4-sphere, called a small exotic . Furthermore we remark that the exoticness of the is connected with the non-trivial smooth h-cobordism , i.e. the failure of the smooth h-cobordism theorem implies the existence of small exotic ’s.
3.3 Exotic and Casson handles
The theorem 3.2 relates a non-trivial h-cobordism between two compact, simple-connected, smooth 4-manifolds to a small exotic . Using theorem 3.1, we can understand where the non-triviality of the h-cobordism comes from: one of the Akbulut corks, say , must be glued in by using a non-trivial involution of the boundary . In the notation above, there is a non-product h-cobordism between and with a h-subcobordism between and . There is an open neighborhood of the h-subcobordism which is an open h-cobordism between the open neighborhoods . Both neighborhoods are homeomorphic to but not diffeomorphic to the standard (as induced from the non-productness of the h-cobordism ). This exotic is the interior of the attachment of a Casson handle to the boundary of the cork .
Now let us consider the basic construction of the Casson handle . Let be a smooth, compact, simple-connected 4-manifold and a (codimension-2) mapping. By using diffeomorphisms of and , one can deform the mapping to get an immersion (i.e. injective differential) generically with only double points (i.e. ) as singularities GolGui:73 (). But to incorporate the generic location of the disk, one is rather interesting in the mapping of a 2-handle induced by from . Then every double point (or self-intersection) of leads to self-plumbings of the 2-handle . A self-plumbing is an identification of with where are disjoint sub-disks of the first factor disk444In complex coordinates the plumbing may be written as or creating either a positive or negative (respectively) double point on the disk (the core).. Consider the pair and produce finitely many self-plumbings away from the attaching region to get a kinky handle where denotes the attaching region of the kinky handle. A kinky handle is a one-stage tower and an -stage tower is an -stage tower union kinky handles where two towers are attached along . Let be and the Casson handle
is the union of towers (with direct limit topology induced from the inclusions ). A Casson handle is specified up to (orientation preserving) diffeomorphism (of pairs) by a labeled finitely-branching tree with base-point *, having all edge paths infinitely extendable away from *. Each edge should be given a label or . Here is the construction: tree . Each vertex corresponds to a kinky handle; the self-plumbing number of that kinky handle equals the number of branches leaving the vertex. The sign on each branch corresponds to the sign of the associated self plumbing. The whole process generates a tree with infinite many levels. In principle, every tree with a finite number of branches per level realizes a corresponding Casson handle. The simplest non-trivial Casson handle is represented by the tree : each level has one branching point with positive sign .
Given a labeled based tree , let us describe a subset of . Now we will construct a which is diffeomorphic to the Casson handle associated to . In embed a ramified Whitehead link with one Whitehead link component for every edge labeled by leaving * and one mirror image Whitehead link component for every edge labeled by (minus) leaving *. Corresponding to each first level node of we have already found a (normally framed) solid torus embedded in . In each of these solid tori embed a ramified Whitehead link, ramified according to the number of and labeled branches leaving that node. We can do that process for every level of . Let the disjoint union of the (closed) solid tori in the th family (one solid torus for each branch at level in ) be denoted by . tells us how to construct an infinite chain of inclusions:
and we define the Whitehead decomposition of . is the Whitehead continuum Whitehead35 () for the simplest unbranched tree. We define to be
alternatively one can also write
where is the cone of a space
over the point . As Freedman (see Fre:82 () Theorem 2.2) showed is diffeomorphic to the Casson handle given by the tree .
3.4 The design of a Casson handle and its foliation
A Casson handle is represented by a labeled finitely-branching tree with base point , having all edge paths infinitely extendable away from . Each edge should be given a label or and each vertex corresponds to a kinky handle where the self-plumbing number of that kinky handle equals the number of branches leaving the vertex. The open handle is represented by the , i.e. there are no kinky handles. One of the cornerstones of Freedmans proof of the homeomorphism between a Casson handle and the open 2-handle are the reembedding theorems. Then one foliates and by copies of the frontier . The frontier of a set is defined by . As example we consider the interior of a disk and obtain for the frontier , i.e. the boundary of the disk . Then Freedman (Fre:82 () p.398) constructs another labeled tree from the tree . There is a base point from which a single edge (called “decimal point”) emerges. The tree is binary: one edge enters and two edges leaving a vertex (or every vertex is trivalent). The edges are named by initial segments of infinite base 3-decimals representing numbers in the standard “middle third” Cantor set555This kind of Cantor set is given by the following construction: Start with the unit Interval and remove from that set the middle third and set Continue in this fashion, where . Then the Cantor set is defined as . With other words, if we using a ternary system (a number system with base 3), then we can write the Cantor set as . . Each edge of carries a label where is an ordered finite disjoint union of 5-stage towers together with an ordered collection of standard loops generating the fundamental group. There is three constraints on the labels which leads to the correspondence between the labeled tree and the (associated) -labeled tree . One calls the design.
Two words are in order for the design : first, every sequence of ’s and ’s is one path in representing one embedded Casson handle where both trees are related like . For example, the Casson handle corresponding to is obtained as the union of the 5-stage towers . For later usage we identify the sequence with the Tree . Secondly, there are gaps, i.e. we have only a Cantor set of Casson handles not a continuum. For instance a gap is lying between the paths and In the proof of Freedman, the gaps are shrunk to a point and one gets the desired homeomorphism. Here we will use this structure to produce a foliation of the design. Every path in is represented by one sequence over the alphabet . Every gap is a sequence containing at least one (so for instance or . There is now a natural order structure given by the sequence (for instance ). The leaves are the corresponding gaps or Casson handles (represented by the union 5-stage towers ending with ). The tree structure of the design should be also reflected in the foliation to represent every path in as a union of 5-stage towers. By the reembedding theorems, the 5-stage towers can be embedded into each other. Then we obtain two foliations of the (topological) open 2-handle : a codimension-1 foliation along one axis labeled by the sequences (for instance ) and a second codimension-1 foliation along the radius of the disk induced by inclusion of the 5-stage towers (for instance ). Especially the exploration of a Casson handle by using the design is given by its frontier, in this case, minus the attaching region. In case of a usual tower we get the frontier with . The gaps have a similar structure. Then the foliation of the Casson handle (induced from the design) is given by the leaves over the disk in the Casson handle, i.e. the disk is foliated by parallel lines (see Fig. 1).
So, every Casson handle with a given tree has a codimension-one foliation given by its design.
This foliation can be also understood as a foliated cobordism. For that purpose we consider the foliation as part of a foliation of the 2-sphere (see Fig. 2).
The 2-sphere is decomposed by , two pole regions ( and an equator region . The foliation of the disk as in Fig. 1 can be used to foliate and . Both foliations can be connected by the leaves which are the longitudes. Then one obtains a foliated cobordism between and given by the obvious foliation of the equator region (a cylinder).
3.5 Capped gropes and its design
In this subsection we discuss a possible generalization of Casson handles. The modern way to the classification of 4-manifolds used “capped gropes”, a mixed variant of Casson handle and grope (chapters 1 to 4 in FreQui:90 ()). We do not want to complicate the situation more than needed. But for later developments we have to discuss some part of the theory but we remark that all results can be easily generalized to capped gropes as well.
A grope is a special pair (2-complex,circle), where the circle is referred to as the boundary of the grope. There is an anomalous case when the depth is : the unique grope of depth 1 is the pair (circle,circle). A grope of depth 2 is a punctured surface with the boundary circle specified (see Fig. 3).
To form a grope of depth , take a punctured surface, , and prescribe a symplectic basis . That is, and are embedded curves in which represent a basis of such that the only intersections among the and occur when and meet in a single point . Now glue gropes of depth along their boundary circles to each and with at least one such added grope being of depth . (Note that we are allowing any added grope to be of depth 1, in which case we are not really adding a grope.) The surface is called the bottom stage of the grope and its boundary is the boundary of the grope. The tips of the grope are those symplectic basis elements of the various punctured surfaces of the grope which do not have gropes of depth attached to them.
A capped grope is a grope with disks (the caps) attached to all its tips. The grope without the caps is sometimes called the body of the capped grope.
The capped grope (as cope) was firstly described by Freedman in 1983Fre:83 (). The caps are only immersed disks like in case of the Casson handle to make the grope simple-connected. The great advantage is the simpler frontier, instead of (see subsection 3.4) one has solid tori as frontier of the capped grope (as shown in AncelStarbird1989 ()). The corresponding design (and its parametrization) can be described similar to the Casson handle by sequences containing and (see section 4.5 in FreQui:90 ()). There are also gaps (described by a in the sequence) who look like .
3.6 The radial family of uncountable-many small exotic
Given a small exotic induced from the non-product h-cobordism between and with Akbulut corks and , respectively. Let be a compact subset. Bizaca and Gompf BizGom:96 () constructed the small exotic by using the simplest tree . Bizaca Biz:94 (); Bizaca1995 () showed that the Casson handle generated by is an exotic Casson handle. Using Theorem 3.2 of DeMichFreedman1992 (), there is a topological radius function (polar coordinates) so that with . Then and is also a small exotic for belonging to a Cantor set . Especially two exotic and are non-diffeomorphic for except for countable many pairs. In DeMichFreedman1992 () it was claimed that there is a smoothly embedded homology 4-disk . The boundary is a homology 3-sphere with a non-trivial representation of its fundamental group into (so cannot be diffeomorphic to a 3-sphere). According to Theorem 3.2 this homology 4-disk must be identified with the Akbulut cork of the non-trivial h-cobordism. The cork is contractable and can be (at least) build by one 1-handle and one 2-handle (case of a Mazur manifold). Given a radial family with radius so that . Suppose there is a diffeomorphism
fixing the compact subset . Then this map induces end-periodic manifolds666We ignore the inclusion for simplicity. and which must be smoothable contradicting a theorem of Taubes Tau:87 (). Therefore and are non-diffeomorphic for (except for countable many possibilities).
3.7 Exotic and codimension-1 foliations
In this subsection we will construct a codimension-one foliation on the boundary of the cork with non-trivial Godbillon-Vey invariant. The strategy of the proof goes like this: we use the foliation of the design of the Casson handle (see subsection 3.4) for the radial family to induce a foliated cobordism . The restriction to its boundary gives cobordant codimension-1 foliations of with non-trivial Godbillon-Vey invariant . In the subsection 2.2 we described a foliation of the 3-sphere unique up to foliated cobordism for every given value of the Godbillon-Vey invariant. By theorem 2.1, we get a corresponding foliation on (with the same Godbillon-Vey invariant). Finally we obtain:
Given a radial family of small exotic with radius and induced from the non-product h-cobordism between and with Akbulut cork and , respectively. The radial family determines a family of codimension-one foliations of with Godbillon-Vey invariant . Furthermore given two exotic spaces and , homeomorphic but non-diffeomorphic to each other (and so ) then the two corresponding codimension-one foliation of are non-cobordant to each other.
The proof can be found in AsselmeyerKrol2009 () and in the appendix A.
In the theorem 3.3 above we constructed a relation between codimension-1 foliations on and the radius for the radial family of small exotic . By using theorem 2.1 we can trace back the foliation on by a foliation on the 3-sphere . This situation can be seen differently by using the diffeomorphism . Then, the foliation on induces a foliation on at least partially. Thus, we have a 3-sphere lying at the boundary of the Akbulut cork inducing a codimension-1 foliation on . Then by theorem 3.3:
Any class in induces a small exotic where lies at the boundary of the cork .
3.8 Integer Godbillon-Vey invariants and flat bundles
Clearly the integer classes are a subset of the full set and one can use the construction above to get the foliation. Especially the polygon must be formed by segments with angles of integer value with respect to to get an integer value for the volume