Big Bang and Topology

In this paper we discuss the initial state of the universe at the Big Bang. By using ideas of Freedman in the proof of the disk embedding theorem for 4-manifolds, we describe the corresponding spacetime as gravitational instanton. The spatial space is a fractal space (wild embedded 3-sphere). Then we construct the quantum state from this fractal space. This quantum state is part of the string algebra of Ocneanu. There is a link to the Jones polynomial and to Witten's topological field theory. Using this link, we are able to determine the physical theory (action) to be the Chern-Simons functional. The gauge fixing of this action determines the foliation of the spacetime and as well the smoothness properties. Finally, we determine the quantum symmetry of the quantum state to be the enveloped Lie algebra $U_{q}(sl_{2}(\mathbb{C}))$ where $q$ is the 4th root of unity.


Introduction
What was the initial state of the universe? This question is fundamental to understand the further development of the universe. However, the usual extrapolation techniques fail here. Therefore, an answer to this question seems to be of global nature. As a mathematical method, therefore, the field of topology would immediately suggest its usefulness here, namely exactly such questions of global nature are answered for spaces within the field. From this motivation we have therefore awakened the construction of spacetime by means of topological methods as a starting point. Here, of course, one can only start with rather general assumptions. All the more astonishing is the result that only a few quite natural assumptions are sufficient to arrive at an unambiguous result. Exactly the discussion of this approach can be found in the next section of the paper. Thereby also the part of spacetime can be identified which belongs to the Big Bang itself. In the search for this part, topology again plays an important role. Special solutions, gravitational instantons,

A (coherent) model for the spacetime
In this section, we will describe the model of the spacetime seen as the space of spacetime events M . At first we start with three (more or less) obvious assumptions to restrict the class of spaces M : smooth 4-manifold (we can use the concept of a differential equation for the dynamics), compactness (every sequence of events is an event) and simply-connectedness (every time-like loop can be contracted to maintain causality at least in principle). Then the spacetime is an open submanifold of M including examples like S 3 ×R. To determine M completely, we need the realization of Ricci-flatness in M representing the vacuum state (no matter) of general relativity. Together with the other assumptions, M is the K3 surface, a Calabi-Yau space of two complex dimensions (i.e. a 4-dimensional real manifold). In the following we will discuss the consequences of this approach. In particular, the K3 surface is a gravitational instanton and using ideas of Hartle and Hawking, the Big Bang can be understood as tunneling event induced from a gravitational instanton. We will argue below that the Big Bang is represented by the 4-disk D 4 with the initial state S 3 . Then the corresponding quantum state must be a fractal space with S 3 −topology. In our previous work, we got a relation between the quantum state and a so-called wildly embedded 3-sphere as fractal space. It is the main result of our argumentation in this section: the initial state of the universe is the fractal 3-sphere. The reader who is willing to accept this assumption can switch to the next section.
There are infinitely many suitable topologies for the spacetime, seen as a 4manifold, and for the space, seen as a 3-manifold. Of course, there are some heuristics but usually not sufficient for determination of the spacetime uniquely. Here we will take a different way. Why not trying to determine the space M of all possible spacetime-events? Therefore we start with a definition: let M be the space of all possible spacetime events, i.e. the set of all spacetime events carrying a manifold structure. Then a specific physical system or configuration is an embedding of a 3-manifold into M and a dynamics is an embedding of a cobordism between 3-manifolds (representing the configuration at the initial and end points) into M . Here, we assume implicitly that everything can be geometrically/topologically expressed as submanifolds (see [15,7]). In the following we will try to discuss this approach and how far one can go. Some heuristic arguments are rather obvious: 1. M is a smooth 4-manifold, 2. any sequence of spacetime event has to converge to a spacetime event and 3. any loop (time-like or not) must be contracted.
A dynamics is known to be a mapping of a spacetime event to a new spacetime event. It is usually a smooth map (differential equations) motivating the first argument. The second argument expresses the fact that any initial spacetime event must converge to a final spacetime event. Or, the limit of any sequence of spacetime events must be converge to a spacetime event. Then, M is a compact, smooth 4-manifold. The usual or actual spacetime is an open subset of M . The third argument above is motivated to neglect time-like loops in principle. If the underlying spacetime is multiple-connected then there are loops in the spacetime which cannot be contracted to a point leading to potential time-like loops. Therefore, a simple-connected spacetime is a necessary condition to avoid closed time-like loops. But, compact spacetime always admit closed time-like loops, see [32]. Therefore, this condition is not sufficient but the usual (or actual) spacetime is an open subset of M or the usual spacetime is embedded in M . Then, if the usual spacetime is also simply-connected then because of the non-compactness, see [32] again, there are no time-like loops. But to understand the property 'simple-connectedness', we consider a loop in the spacetime. If this loop cannot contract then there are two ways or two different curves connecting two different events. By changing the embedding of the curves via a diffeomorphism (this procedure is called isotopy), we can deform one curve to agree with other curve. Or, every loop formed by the two curves can be contracted. Therefore this argument implies that there are no timelike loops and the non-compactness of the open subset implies causality. Finally, M is a compact, simply connected, smooth 4-manifold.
The following restrictions of M will determine the spacetime completely. For that reason we demand that the equations of general relativity are valid without any restrictions. Then the vacuum equations are given by R µν = 0 so that we get Ricci-flatness. But as shown in [20,19] and in recent years in [15,7,3], the coupling to matter can be described by a change of the smoothness structure. Therefore the modification of the smoothness structure will produce matter (or sources of gravity). But at the same time, we need a smoothness structure which can be interpreted as vacuum given by a Ricci-flat metric. Therefore we will demand that 4. M has to admit a smoothness structure with Ricci-flat metric representing the vacuum.
Interestingly, these four demands are restrictive enough to determine the topology of M completely. With the help of Yau's seminal work [50], that the K3 surface is the unique compact, simply connected Ricci-flat 4-manifold, we will obtain that M is topologically equivalent (homeomorphic) to the K3 surface. But it is known by the work of LeBrun [41] that there are non-Ricci-flat smoothness structures. Therefore in a next step, we will determine the smoothness structure of M . For that purpose, we will present some known results in differential topology of 4-manifolds (see [6] for details and the construction of the E 8 −manifold): • there is a compact, contractible submanifold A ⊂ M (called Akbulut cork) so that cutting out A and reglue it (by an involution) will produce a new smoothness structure, • M splits topologically into two copies of the E 8 −manifold and three copies of S 2 × S 2 and • the 3-sphere S 3 is a submanifold of A.
In [9] we already discussed this case. From the topological point of view, any sum of E 8 −manifolds and S 2 × S 2 is realized by a closed, simply-connected, topological 4-manifold but not all topological 4-manifolds are smooth manifolds. To clarify this point, let us consider the 4-manifold which splits topologically into p copies of the |E 8 | manifold and q copies of S 2 × S 2 or Then, this 4-manifold is smoothable for every q but p = 0. The first combination for p = 0 is the pair of numbers p = 2, q = 3 (which is the K3 surface). Any other combination (p = 2, q < 3 or every q and p = 1) is forbidden as shown by Donaldson [25]. Therefore the simplest combination of |E 8 | and S 2 × S 2 is realized by the K3 surface. Now we consider the smooth K3 surface which is Ricci-flat, simply connected and smooth. A main part in the following discussion will be the usage of the smoothness condition. As discussed above, the smoothness structure is determined by the Akbulut cork A. Furthermore as argued above, the smoothness structure is strongly related to the appearance of matter (see [15,7,3]) and this process is strongly connected to the evolution of our cosmos (see [12,14]). This process is known as reheating after the inflationary phase. Therefore, the Akbulut cork (including its embedding) should represent the inflationary phase with reheating. We have already discussed this partly in our works (see [13,3] for first results in this direction).
The central submanifold determining the smoothness structure is the Akbulut cork A, a contractible submanifold with boundary ∂A. As shown by Freedman [27], the Akbulut cork is build from a homology 3-sphere which will become the boundary ∂A. The difference to a usual 3-sphere S 3 is given by the so-called fundamental group, the equivalence class of closed loops up to deformation (homotopy) with concatenation as group operation. In principal, one constructs a cobordism between S 3 and the homology 3-sphere ∂A. All elements of the fundamental group will be killed by adding appropriate disks. At the end, one can add a 4-disk to get the full contractible cork A. The topology of ∂A depends strongly on the topology of M . In case of the K3 surface, ∂A is known to be a Brieskorn spheres, precisely the 3-manifold ∂A = Σ(2, 5, 7) = x, y, z ∈ C | x 2 + y 5 + z 7 = 0|x| 2 + |y| 2 + |z| 2 = 1 .
The construction of the smoothness structures is based on the work [16,17]. The smoothness structure depends on the Casson handle (used to construct an exotic R 4 in the cited work). A Casson handle is uniquely determined by a branched tree. Then the simplest Casson handle is given by an unbranched tree and we will choose this smoothness structure in the following. The corresponding K3 surface is constructed in [17].
The embedding of the Akbulut cork is essential for the following results. In [12] it was shown that the embedded cork admits a hyperbolic geometry if the underlying K3 surface has an exotic smoothness structure. Additionally the open neighborhood N (A) of the Akbulut cork in the K3 surface is an exotic R 4 , i.e. a space homeomorphic to the Euclidean space R 4 but not diffeomorphic to it. In the following we will denote this exotic R 4 as R 4 . One of the characterizing properties of an exotic R 4 (all known examples) is the existence of a compact subset K ⊂ R 4 which cannot be surrounded by any smoothly embedded 3-sphere (and homology 3-sphere bounding a contractible, smooth 4-manifold). But there is always a topologically embedded 3-sphere i.e. this 3-sphere is wildly embedded. In [3] we described this wildly embedded 3-sphere explicitly (denoted as Y ∞ ) and we showed in [11] that this wildly embedded 3-sphere can be understood as quantum state i.e. it is the deformation quantization of a tame (or usual) embedding. The notation wildly embedded or wild is purely mathematical. Instead we will denote this wild 3-sphere as fractal 3-sphere. But at first we will look at the Akbulut cork A which can be decomposed as where W (S 3 , ∂A) describes a cobordism between the 3-sphere and the boundary ∂A = Σ(2, 5, 7). In [12] we discussed this cobordism W (S 3 , ∂A) as the first (inflationary) transition S 3 → ∂A from the initial state (the 3-sphere) to a non-trivial space (containing matter). Then by using the embedding of A into the K3 surface, we identify the 3-sphere (boundary of D 4 ) with the wild 3-sphere Y ∞ (from the open neighborhood N (A)), or the initial state of our model of the universe is a fractal 3-sphere (which is a quantum state, see [11,3]). With this identification in mind, we are able to interpret the first transition W (S 3 , ∂A) (from the wild 3-sphere to the (classical) non-trivial state ∂A) as decoherence process, see [10]. In [12], we discussed a second transition leading to a cosmological constant. Finally we have the two transitions where P denotes the Poincare sphere. In this paper we are interested in the formation of the initial state (the fractal 3-sphere), also called Big Bang. Using the decomposition (2), this formation is expressed in spacetime via the 4-manifold D 4 having the boundary ∂D 4 = S 3 , the (fractal) 3-sphere. Again, the embedding of D 4 into the K3 surface is important, otherwise one will never obtain the fractal 3-sphere as boundary. Therefore, many properties of the K3 surface go over to D 4 by using the embedding.
To describe this embedding, we need the following fact: the K3 surface is a gravitational instanton. We implicitly used this fact above when we constructed a simplyconnected, Ricci-flat spacetime (uniquely given by the K3 surface). In general, an instanton is a field configuration, which is interpreted as a tunneling effect between topologically in-equivalent sectors of the vacuum. The term "gravitational instanton" is usually used for 4-manifolds whose Weyl tensor is self-dual and fulfills the Einstein condition Ri c = Λg . Usually it is assumed that the metric is asymptotic to the standard metric of Euclidean 4-space. In case of the K3 surface, there is the phenomenon that gravitational instantons are created by bubbling off a subspace. Here we recommend the recent publication [35] for the description of this process. To state it more precisely, there is a family of hyperkähler metrics g β on a K3 surface which collapse to an interval [0, 1] in the Gromov-Hausdorff limit (β → ∞ with metrics d t 2 ) with Taub-NUT bubbles in the interior and Tian-Yau metrics at the endpoints. For the embedding of D 4 we choose the Taub-NUT metric in the (open) neighborhood of the boundary. But what about the interior of D 4 ? Here, we have to use the elliptic fibration of the K3 surface (as torus bundle over the S 2 with singular fibers, see [30]). Then we can describe the embedded D 4 by the Eguchi-Hanson metrics (a gravitational instanton). This metric is a Riemannian metrics. Here, the signature of the metrics changes from the Riemannian signature (for D 4 ) to the Lorentzian signature (for ∂D 4 × (0, 1)). In a recent publication [36], a gravitational instanton with these properties is constructed. The construction used explic-itly the hyperkähler structure (SU (2) holonomy group). The gluing of the instanton solutions can be done by using the work [35].
As explained above, the boundary ∂D 4 is identified with the wild (or fractal) 3sphere. Then the signature change of the metric can be identified with the formation of this fractal 3-sphere. Here we follow the usual interpretation (Hartle-Hawking and Hawking-Turok see [33,34]) that the gravitational instanton D 4 represents the Big Bang (via a tunneling event) leading to the quantum state of the universe. In [11] we showed that a quantum state can be topologically understood as a wildly embedded 3-sphere or a fractal 3-sphere for short. Therefore we will argue accordingly that the quantum state of the universe (as initial state) is represented by the fractal 3-sphere.
In the next section we will describe this fractal 3-sphere explicitly.

The construction of the fractal 3-sphere as quantum state
In [12,14,13,4], we described a model for the cosmic evolution which is in good agreement to current measurements [1,2]. Amazingly as discussed above, we are able to extrapolate the state at the Big Bang [3,4]: a fractal 3-sphere as boundary of a 4-disk D 4 , i.e. a gravitational instanton as transition (tunneling) to a fractal 3sphere representing the quantum state [11]. Furthermore as explained above, this fractal 3-sphere is part of R 4 , an exotic R 4 . Before we start with the construction of the fractal 3-sphere, we will describe the physical ideas behind the construction. In the introduction, we explained the concept of a wild embedding (or fractal space). In short, a wild embedding is a submanifold (image of an embedding map) which must be decomposed into infinitely many substructures (polygons etc.). Therefore it contains an infinite amount of information. In our previous work we showed that the wild embedding is an expression for a quantized geometry. In case of a fractal 3-sphere (as wildly embedded 3-sphere), one decomposes the 3-sphere into similar looking pieces with constant curvature. Every piece has a different curvature so that the whole fractal 3-sphere represents the set of possible curvatures. These structures appear at all scales. Because of this property, we have to use the methods of noncommutative geometry to get a rigorous definition of this procedure. The following construction of the fractal 3-sphere is directly motivated by the exotic smoothness structure. The basic structure is a tree (used to define the Casson handle). Every part of the tree like edge or vertex is associated to a 3-manifold. For the whole tree, one gets an infinitely complicated 3-manifold which is topologically equivalent to a 3-sphere. This fractal 3-sphere is the boundary of a 4-disk or 4-ball described in the next section and representing the Big Bang as gravitational instanton (via a tunneling event).
In [3] we described this fractal 3-sphere as a sequence of 3-manifolds with increasing complexity. At first we want to comment on the uniqueness of the construction. The sequence of 3-manifolds is determined by the smoothness structure or better by the Casson handle which is used to construct this structure. Every Casson handle is represented by a tree. This tree is translated into a link: every nbranching point (vertex of the tree) is given by a Whitehead link with n circles and every line (edge of the tree) is given by the circle of the Whitehead link. In the previous section, we introduced the smoothness structure as given by the unbranched tree. Obviously, the unbranched tree is a subtree for any other more complex tree. It is a fundamental property of Casson handles (see [27]) that a Casson handle C H 1 embeds into another Casson handle C H 2 , say C H 1 ⊂ C H 2 , iff the tree of C H 2 embeds into the tree of C H 1 . Therefore any other Casson handle embeds into the Casson handle represented by the unbranched tree. This property is unique for the smoothness structure and the construction of the fractal 3-sphere.
For completeness we will shortly explain the construction. The 3-manifold Y 0 is given by surgery (0−framed) along the pretzel knot (−3, 3, −3) (or the knot 9 46 in Rolfson notation), Y 1 is constructed by 0−framed surgery along the Whitehead double of the pretzel knot (−3, 3, −3) and finally Y n is constructed by 0−framed surgery along the nth Whitehead double of the pretzel knot (−3, 3, −3). In the limit n → ∞, we obtained Y ∞ as 0−framed surgery along the ∞th Whitehead double of the pretzel knot (−3, 3, −3) (a so-called wild knot). This 3-manifold Y ∞ is the fractal 3-sphere (it has the topology of a 3-sphere by a theorem of Freedman [27]). The whole process can be seen as an iteration process at the level of 3-manifolds: we start with Y 0 and end with Y ∞ , the fractal 3-sphere.
To understand this abstract construction (via Dehn surgery or Kirby calculus [30]) we have to describe the construction of the first 3-manifold Y 0 more carefully. For that purpose, we have to describe Dehn surgery or surgery along a knot. If we remove a thicken knot N (K ) = K × D 2 (so-called tubular neighborhood) from the 3-sphere S 3 then one obtains the knot complement C (K ) = S 3 \ N (K ). Now we glue in one solid torus D 2 × S 1 to C (K ) by a mapping of the boundary φ : ∂C (K ) = T 2 → ∂(D 2 × S 1 ) = T 2 so that we get All closed curves on a torus can be generated by the two possible non-contracting curves m, ℓ the meridian and longitude, respectively. In principle, any closed curve γ on a torus T 2 is given by two numbers with [γ] = [aℓ + bm] (for the homotopy classes). Then the map φ is characterized by a mapping of the meridian m of one torus to the curve γ determined by the ratio r = b/a (including ∞ for a = 0) called the frame number. As a warm-up example, we consider the 0-framed surgery along the unknot S 1 in S 3 . The knot complement of the unknot C (S 1 ) = D 2 × S 1 is glued to another solid torus D 2 × S 1 (along its boundary ∂(D 2 × S 1 ) = S 1 × S 1 ) with framing 0 which means that the meridian of ∂C (S 1 ) is mapped to the meridian of ∂(D 2 × S 1 ). But that means that D 2 ×S 1 is glued to D 2 ×S 1 along the boundary, i.e. (D 2 ∪ ∂D 2 D 2 )× S 1 = S 2 × S 1 . Therefore 0−framed surgery along the unknot gives S 2 × S 1 . Interestingly, 0−framed surgery along any knot produces a 3-manifold which is very similar to S 2 × S 1 (having the same homology). Every Y n in the sequence above is produced by 0−framed surgery along an knot of increasing complexity. One starts for n = 0 with the knot 9 46 (in Rolfson notation) producing Y 0 then n = 1 with Y 1 is produced by the Whitehead double W h 1 (9 46 ) of this knot, Y 2 is given by the second iterated Whitehead double W h 2 (9 46 ) and so on. In the limit n → ∞ one gets Y ∞ as 0−framed surgery along the ∞−iterated Whitehead double W h ∞ (9 46 ) of 9 46 (a so-called wild knot). But this limit changes the topology of Y ∞ . For every finite n ≥ 0, Y n has the same homology as S 2 × S 1 but Y ∞ is topologically equivalent to S 3 (by a theorem of Freedman [27]).
In [11,3] we constructed a quantum state from a wild embedding. Main idea is the description of the wild embedding by using operator algebras in the spirit of noncommutative geometry. This relation is strict: the wild embedding is in one-toone relation to a foliation with leaf space a factor III von Neumann algebra known as the observable algebra of a quantum field theory. To understand this relation from a geometrical point of view, we will use the decomposition of the factor III into a factor II and a one-parameter group of automorphisms. We remark that this decomposition was used by Rovelli and Connes [24] to introduce a time variable in quantum gravity. This decomposition means that in some sense the intractable factor III can be reduced to the easier accessible factor II (operators of finite trace).
For completeness we will also present the construction (see [11]) of the C * −algebra from the wild embedded 3-sphere. Let I : S 3 → R 4 be a wild embedding of codimensionone so that I (S 3 ) = S 3 ∞ = Y T . Now we consider the complement R 4 \ I (S 3 ) which is non-trivial, i.e. π 1 (R 4 \ I (S 3 )) = π = 1. Now we define the C * −algebra C * (G , π) associated to the complement G = R 4 \ I (S 3 ) with group π = π 1 (G ). If π is non-trivial then this group is not finitely generated. From an abstract point of view, we have a decomposition of G by an infinite union of 'level sets' C i . Then every element γ ∈ π lies (up to homotopy) in a finite union of levels.
The basic elements of the C * −algebra C * (G , π) are smooth half-densities with compact supports on G , f ∈ C ∞ c (G , Ω 1/2 ), where Ω 1/2 γ for γ ∈ π is the one-dimensional complex vector space of maps from the exterior power Λ k L (dimL = k), of the union of levels L representing γ, to C such that For f , g ∈ C ∞ c (G , Ω 1/2 ), the convolution product f * g is given by the equality with the group operation γ 1 •γ 2 in π. Then we define via f * (γ) = f (γ −1 ) a * operation making C ∞ c (G , Ω 1/2 ) into a * algebra. Each level set C i consists of simple pieces (in case of Alexanders horned sphere, we will explain it below) denoted by T . For these pieces, one has a natural representation of C ∞ c (G , Ω 1/2 ) on the L 2 space over T . Then one defines the representation The completion of C ∞ c (G , Ω 1/2 ) with respect to the norm π). Finally we are able to define the C * −algebra associated to the wild embedding. Using a result in [11], one can show that the corresponding von Neumann algebra is the factor III 1 . This algebra is the observable algebra of a free (algebraic) quantum field theory with one vacuum vector [18]. Here we will discuss an alternative way to construct the factor III 1 . For that purpose, we look again at the construction of the wild 3-sphere Y ∞ . The ∞−iterated Whitehead double W h ∞ (9 46 ) of the knot 9 46 gives a wild knot K and Y ∞ can be constructed by the 0−framed surgery. In [11], we discussed the known result that the (deformation) quantization of the geometric structures (space of constant curvature) is given by the Kauffman bracket skein module. For Y ∞ it means that we have to consider the Kauffman bracket skein module K h (C (K )) of C (K ). Here, it is known that K h (C (K )) is a module over the noncommutative torus which is related (for h = 0) to the boundary ∂C (K ) = T 2 . The noncommutative torus defines a factor II ∞ algebra and we will show in our forthcoming work that the whole K h (C (K )) gives the factor III 1 .

The quantum spacetime at the Big Bang
In section 2 we described the Big Bang as gravitational instanton D 4 (induced from spacetime, the K3 gravitational instanton). The initial state of the universe is given as the boundary ∂D 4 = S 3 , a wild 3-sphere, via a tunneling process (Hartle-Hawking). Usually, nothing is known about the formation of the initial state via the tunneling process. In contrast, we have here the comfortable situation that there is a relation between the boundary -the wild 3-sphere -and the interior of the 4-disk. There is a process for the formation of the wild 3-sphere, which is divided into an infinite number of subprocesses, called Casson handles. This structure is called the design and was developed for the classification of 4-manifolds [27,26]. All subprocesses can be parameterized by all paths in a binary tree. The detailed construction of these Casson handles is unimportant for the following (but see [27]). Again before we start with the construction, we will discuss the physics behind it. Like in case of the fractal 3-sphere, the design is a geometric/topological expression for the quantum state of the spacetime. Here, it is the formation of the fractal 3-sphere seen as the boundary of the 4-disk. The design is a summation over all possible formation processes. It is an expression for the functional integral. Like for the construction of the fractal 3-sphere, we also get complicated substructures at all scales so that we need the methods of the noncommutative geometry again. Here, the formation process is parametrized by a binary tree where every path is a particular process. But we need all processes or pathes of the binary. Therefore we associate to every path an operator which consists of a sum of elementary operators (projection operators). Then one gets directly an operator algebra (Temperley-Lieb algebra) which can be interpreted as an algebra of field operators. Here, we use the fact that we consider paths of a binary tree: the operator algebra is the algebra of fermion field operators. Interestingly, the expectation value in this algebra is related to a structure (Jones polynomial) which is well-known for 3-dimensional manifolds and knots. Now we argue backwards: the expectation value is defined by a functional integral with Chern-Simons action in agreement with our previous work. The Chern-Simons action in the light cone gauge is interpreted as invariant of the underlying foliation of the spacetime. Again with the help of noncommutative geometry, we are able to get a kind of quantum action (the so-called flow of weigths). We remark that at the topological level we have a kind of duality between the design (4D) and links (3D) which will be further investigated in our forthcoming work.
The design S(Q) is a structure to label all Casson handle which embed in a given Casson handle Q. In our case, this Casson handle Q is represented by an unbranched tree. Then this Casson handle Q represents (in some sense) all Casson handles. We will define this design S(Q) to be the quantum state of Q. Below we will determine the operator algebra associated to Q and we will show that this algebra is a von Neumann algebra of finite trace as well with one vacuum vector (factor II 1 ). But at first we will describe the construction of the design S(Q). In [8] we also described this construction but in a different context. For completeness we will present this construction again.
According to Freedman ([27] p.393), a Casson handle is represented by a labeled finitely-branching tree Q with basepoint ⋆, having all edge paths infinitely extendable away from ⋆. Each edge should be given a label + or − . The tree Q is fixed generating the wild 3-sphere (as the boundary of D 4 ). Then Freedman ([27] p.398) constructs another labeled tree S(Q) from the tree Q. 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. The edges are named by initial segments of infinite base 3-decimals representing numbers in the standard "middle third" Cantor set C .s. ⊂ [0, 1]. This kind of Cantor set is given by the following construction: Start with the unit Interval S 0 = [0, 1] and remove from that set the middle third and set S 1 = S 0 \ (1/3, 2/3) Continue in this fashion, where S n+1 = S n \{middle thirds of subintervals of S n }. Then the Cantor set C .s. is defined as C .s. = ∩ n S n . With other words, if we using a ternary system (a number system with base 3), then we can write the Cantor set as C .s. = {x : x = (0.a 1 a 2 a 3 . . .) where each a i = 0 or 2}. Each edge e of S(Q) carries a label τ e where τ e is an ordered finite disjoint union of 6-level-subtrees. There is three constraints on the labels which leads to the correspondence between the ± labeled tree Q and the (associated) τ-labeled tree S(Q).
Every path in S(Q) represents one tree leading to a Casson handle. Any subtree represents a Casson-handle which embeds in Q, see above. Now we will introduce an (operator) algebra structure on S(Q). For that purpose, we have to consider pairs of paths in the (dual) tree of S(Q). Thus we have to concentrate on the so-called string algebra according to Ocneanu [45]. For that purpose we define a non-negative function µ : E d g es → C together with the adjacency matrix △ acting on µ by where s(v) and r (v) denote the source and the range of an edge v. A path in the tree is a succession of edges ξ = (v 1 , v 2 , . . . , v n ) where r (v i ) = s(v i+1 ) and we writeṽ for the edge v with the reversed orientation. Then, a string on the tree is a pair of paths ρ = (ρ + , ρ − ), with s(ρ + ) = s(ρ − ), r (ρ + ) ∼ r (ρ − ) which means that r (ρ + ) and r (ρ − ) ending on the same level in the tree and ρ + , ρ − have equal lengths i.e. |ρ + | = |ρ − | expressing the previous described property r (ρ + ) ∼ r (ρ − ) too. Now we define an algebra St r i ng (n) with the linear basis of the n-strings, i.e. strings with length n and the additional operations: where · can be seen as the concatenation of paths. We normalize the function µ by µ(r oot ) = 1. Now we choose a function µ in such a manner that for a complex number β. Then we can construct elements e n in the algebra St r i ng (n+1) by which are the generators of the so-called Temperley-Lieb algebra. A Temperley-Lieb algebra is an algebra with unit element 1 over a number field K generated by a countable set of generators {e 1 , e 2 , . . .} with the defining relations e 2 i = τ · e i , e i e j = e j e i : |i − j | > 1, e i e i+1 e i = τe i , e i+1 e i e i+1 = τe i+1 , e * i = e i (6) where τ is a real number in (0, 1]. By [38], the Temperley-Lieb algebra has a uniquely defined trace Tr which is normalized to lie in the interval [0, 1]. The generators (5) also fulfill these algebraic relations (6) where τ = β −2 . The trace of the string algebra given by and defines on A ∞ = ( n St r i ng (n) , t r ) an inner product by 〈x, y〉 = t r (x y * ) giving after completion the Hilbert space L 2 (A ∞ , t r ). Now we will determine the parameter τ. Originally, Ocneanu introduce its string algebra to classify the splittings of modules over an operator algebra (see also [31]). Thus, to determine this parameter we look for the simplest generating structure in the tree. The simplest structure in the binary tree S(Q) is one edge which is connected with two other edges. This graph is represented by the following adjacency matrix   0 1 1 1 0 0 1 0 0   having eigenvalues 0, 2, − 2. According to our definition above, β is given by the greatest eigenvalue of this adjacency matrix, i.e. β = 2 and thus τ = β −2 = 1 2 . Then, without proof, we state that the algebra R is given by the Clifford algebra on R ∞ . The coefficients of this algebra are given by a map µ : E d g es → C.
The definition of the trace (7) (or better the inner product) has a strong link to knot theory. This algebra (6) was used by Jones [38,39] to define a new knot invariant. Therefore, we can interpret every expectation value as the knot/link invariant of a certain knot/link (represented by a braid) or a sum of these invariants. But before we have to map the projectors e i to the generators g i so that e k = 1 1+i + 1 2i g k (for the special value τ = 1 2 ), see [39]. Then every generator b i of the braid group B n is mapped to g i (and vice versa). So, the expectation value is associated to a (formal sum) of braids. The closure of these braids are links or every string ρ defines a (formal) sum of links L ρ . Then t r (ρ) must be equal (by definition) to the Jones polynomial V L ρ (t = i ) for the link L ρ for the special value t = i (in general t = exp(i πτ)). The value of the Jones polynomial for t = i is known to be where ℓ is the number of components for L ρ and Ar f (L) is the Arf-invariant of the link (see [44] for the proof of the result and the definitions).
By this chain of arguments, we are able to derive a further link to understand the underlying action for calculating the expectation value. In [49], Witten constructed a topological quantum field theory (TQFT) for the Jones polynomial. This theory has its home on a 3-manifold Σ and we will discuss below this 3-manifold. Let A be a connection of a SU (2) principal bundle over Σ. The Chern-Simons action is given by then from [49] one has the relation between the trace t r (ρ) and the functional integral over the action (8) where W A (L) is the Wilson loop along the link L for the connection A. With this trick, we get the action functional (Chern-Simons action) and the observable (Wilson loop) for the underlying physical theory. The Jones polynomial is known to be intricately connected with the quantum enveloping algebra of the Lie algebra of the group SL(2, C), see [46]. In our case, the parameter q = t = i is the 4th root of unity and it is known that this quantum q-deformation of the Lie algebra sl 2 (C) yields a finite dimensional modular Hopf algebra. Therefore we have determined the underlying quantum symmetry (of the initial state at the Big Bang) as the enveloped algebra U q (sl 2 (C)). Furthermore in [49], a relation between the (2 + 1)−dimensional Chern-Simons theory and a (1 + 1)−dimensional conformal field theory is also discussed.
In particular it was shown that the Hilbert space of pure Chern Simons theories is isomorphic to the space of conformal blocks of an underlying Conformal Field Theory. This link seem to imply that there is an underlying (1 + 1)−dimensional theory. We discussed a similar mechanism in [4] using the Morgan-Shalen compactification and will study the relation between the two approaches in our forthcoming work. Now we have to determine the 3-manifold Σ in the definition of the Chern-Simons theory. At the first view, we identify Σ with the wild 3-sphere. Then this theory is stationary, i.e. it contains no time variable. But as explained above, the formation of the wild 3-sphere can be seen as a process where the 3-manifold is growing by attaching 3-dimensional pieces along surfaces. In the definition of the string algebra, we used Casson handles to define the generators e i . But Casson handles have an inherent 2-dimensional definition (neighborhood of immersed disks) which is used to define the construction of the wild 3-sphere (see [3] for a detailed construction). Then we can see the 3-manifold Σ as a non-trivial cobordism between surfaces (used to define the wild 3-sphere), i.e. we define the Chern-Simons theory as a (2 + 1)−dimensional theory right in the sense of Witten [49]. The 3-manifold is foliated by the surfaces. To construct this foliation, we introduce light cone coordinates (x + = x 0 + x 1 , x − = x 0 − x 1 , x 2 ) together with the connection 1-form (following [40] sec. 4). Now we choose the gauge A − = 0 (axial gauge) so that we have a non-zero gauge field for the future light cone (seen from the Big Bang). Then the Chern-Simons action simplifies to and the restriction of the SU (2) bundle to the surface leads to a bundle reduction from SU (2) to U (1) bundle with an abelian connection a and Chern-Simons form This form has a different interpretation in foliation theory: it is the Godbillon-Vey invariant [29]. Recall that a foliation (M, F ) of a manifold M is an integrable subbundle As shown in [37,47], foliations are really complicate. In the language of noncommutative geometry, the leaf space of a foliation with non-vanishing Godbillon-Vey invariants is a von-Neumann algebra which contains a factor III subalgebra. As shown by Connes [22,23], the Godbillon-Vey class GV can be expressed as cyclic cohomology class (the so-called flow of weights) of the C * −algebra for the foliation. Then we define an expression S = Tr ω (GV HC ) uniquely associated to the foliation (Tr ω is the Dixmier trace). The expression S generates the action on the factor by so that S is the action or the Hamiltonian multiplied by the time. We have evaluated this expression for some cases in [3] and we interpret it as quantum action. A detailed analysis will be shifted to our forthcoming work. But this action is partly satisfactory. In noncommutative geometry, one introduces a spectral triple with a Dirac operator as main ingredient. So, let us consider a Dirac operator D Σ on Σ. As a second ingredient, we introduced a codimension-1 foliation along the 1-form a which is interpreted as abelian gauge field. To take this foliation into account, we couple the abelian gauge field a and the spinor ψ to the Dirac-Chern-Simons action functional on the 3-manifold with the critical points at the solution where τ(ψ, ψ) is the unique quadratic form for the spinors locally given byψγ µ ψ. Now we consider a spacetime Σ × I , so that the solution is translationally invariant. Expressed differently, we choose a spacetime with foliation induced by the foliation of Σ extended by a translation. An alternative description for this choice is by considering the gradient flow of these equations But it is known that this system is equivalent to the Seiberg-Witten equation for Σ×I by using an appropriate choice of the Spi n C structure [42,43]. Then this Spi n C structure is directly related to the foliation. Therefore a non-trivial foliation together with a spectral triple (Dirac operator) induce a non-trivial solution of the gradient system which results in a non-trivial solution of the Seiberg-Witten equations. But this non-trivial solution (i.e. ψ = 0, a = 0) is a necessary condition for the existence of an exotic smoothness structure. So we have a closed circle: we started with a smooth spacetime at the Big Bang forming the initial state. If this state is a wild 3-sphere, we get a non-trivial foliation (=non-vanishing Godbillon-Vey invariant) which produces a non-trivial solution of the Seiberg-Witten equations. Before closing this section, we will discuss the dynamical interpretation of the string algebra above and the observable. The design S(Q) relative to a Casson handle Q (in our case the unbranched tree) is the sum over all Casson handles leading to the quantum state (the fractal 3-sphere as constructed from Q). The string algebra for the binary tree (representing the design) is the Clifford algebra of the Hilbert space. From the physics point of view, it is the algebra of fermion field operators. Every field operator is given by a path in the binary tree (weighted by some coefficients). A combination of the results in [7] and [8] showed that the fermion field operators (as elements of the Ocneanu string algebra) can be also interpreted as the leaf space of a type III foliation (see [37]) seen as a crossed product of the string algebra and its modular automorphism group. This product with the automorphism group is timedependent representation of the field operators (see [24]). Therefore, the foliation of type III (having a non-zero Godbillon-Vey invariant) is the dynamical interpretation of string algebra. But we know more because the design was seen as the formation of the fractal 3-sphere as given by a sequence of 3-manifolds. This process is given by a sequence of 3-manifold topology change which was described in [13]. It leads to an inflationary behavior which is approximately described by a de Sitter space (see [12]). In [21] the algebra of observable for a de Sitter space is described to be a von Neumann algebra of type II 1 . Here we conjecture that there must be a relation between our string algebra and this algebra of observables.

Conclusion
In this paper we have worked out a model of the Big Bang driven by topological considerations. The starting point was the construction of a spacetime as a global expression of the evolution of the universe. But the real core of the paper is the construction of the initial state as a wildly embedded or fractal 3-sphere. Here the construction of a corresponding operator algebra was the decisive step to understand this state. Here many interrelations to other theories came to light. Thus, the expectation value in the operator algebra can always be reinterpreted as a knot invariant (Jones polynomial). The action of the theory is the Chern-Simons invariant, which already appears in the description of a (2+1)-dimensional gravitational theory. In general, these relations to conformal field and Seiberg-Witten theory are the real strength of this approach. This work only prepares the ground for further approaches to the understanding of the initial state at the Big Bang. In our next work we will interpret and calculate the dark matter density as a topological quantity.

A Appendix
In this appendix we will describe the methods and results in [11] to make the paper as self-contained as possible. There, we showed that the deformation quantization of a tame embedding is a wild embedding.
At first we start with some definitions. A map f : N → M between two topological manifolds is an embedding if N and f (N ) ⊂ M are homeomorphic to each other. An embedding i : N → M is tame if i (N ) is represented by a finite polyhedron homeomorphic to N . Otherwise we call the embedding wild. Let I : K n → R n+k be a wild embedding of codimension k with k = 0, 1, 2. Now we assume that the complement R n+k \ I (K n ) is non-trivial, i.e. π 1 (R n+k \ I (K n )) = π = 1. Wild embedding are usually characterized by this property, but if the group π 1 (R n+k \I (K n )) = 1 is trivial then the group π 1 (I (K n )) must be non-trivial for wild embeddings. In section 3 we defined the C * −algebra C * (G , π) associated to the complement G = R n+k \ I (K n ) with group π = π 1 (G ). Therefore the methods of noncommutative geometry are applicable.
For the relation between the tame and wild embedding, we consider the space of geometric structures on the embedded manifold. In [11] we do the calculations for Alexanders horned sphere. The space of geometric structures with isometry group SL(2, C) admits a Poisson structure. The deformation quantization of this Poisson structure is known as Drinfeld-Turaev quantization. In a series of papers it was shown that the deformation quantization of the space of geometric structures with isometry group SL(2, C) is the Kauffman bracket skein algebra. In case of Alexanders horned sphere, we showed that the Kauffman bracket skein algebra is the factor II 1 algebra isomorphic to the enveloping von Neumann algebra of the C * algebra defined by the wild embedding. In particular for a tame embedding, the skein algebra is trivial (it is only a 1-dimensional algebra, the center).