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We propose a model with 3-dimensional spatial sections, constructed from hyperbolic cusp space glued to Seifert manifolds which are in this case homology spheres. The topological part of this research is based on Thurston’s conjecture which states that any 3-dimensional manifold has a canonical decomposition into parts, each of which has a particular geometric structure. In our case, each part is either a Seifert fibered or a cusp hyperbolic space. In our construction we remove tubular neighbourhoods of singular orbits in areas of Seifert fibered manifolds using a splice operation and replace each with a cusp hyperbolic space. We thus achieve elimination of all singularities, which appear in the standard-like cosmological models, replacing them by “a torus to infinity”. From this construction, we propose an alternative manifold for cosmology with finite volume and without Friedmann-like singularities. This manifold was used for calculating coupling constants. Obtaining in this way a theoretical explanation for fundamental forces is at least in the sense of the hierarchy.

In his field equations, Einstein just provided a compact mathematical tool that made it possible to develop a theory to describe the general configuration of matter-energy and space-time for the Universe. Nevertheless, the questions related to the global shape of space and, in particular, its finite or infinite extension, cannot be fully answered by the General Relativity (a local physical theory). That is why, the use of topology (a global mathematical theory) is necessary to construct a new manifold which could answer some of those questions. Accordingly, many topological “alternatives” of three-dimensional spaces can thus be used to build such cosmological universe models.

The investigation of topological changes may bring a new approach to studying the early cosmology of the universe [

Efremov [

We like to point out that our Model of Space-Time does not have singular orbits (those are obtained by factorization with respect to the action group

Other works which can be a motivation for using hyperbolic manifolds in physics are: [

First we make the decomposition of Seifert Fibered Spaces in prime spaces. This process includes a collection of cuts of a 3-dimensional space

After that, we involve the decomposition of cuts along a torus. The reverse operation to splice described in [

Here, we extend those ideas using Thurston’s conjecture (now proved) [

There are several definitions in literature about Seifert Manifolds, here we introduce our own definition of Seifert Manifolds. For hyperbolic spaces we will use Thurston’s theory. Then we will use all those concepts for the construction of a new Manifold representing the cosmological Universe.

Seifert ManifoldsDefinition 1. A Seifert Fibered Manifold (SFM) is an orientable three-manifold

1)

2)

3) The covering translation group is generated by

Definition 2. If

is commutative.

Applying the previous definitions. Consider the trivial fibered cylinder

The reader can also see [

See Appendix for other useful definitions.

In his notes, Thurston [

The upper half-space model (taked from [

This is closely related to the Poincaré disc model and the southern hemisphere, but it is often more convenient for computation or for constructing graphics. To obtain the upper-half space model, rotate the sphere

A hyperbolic line, in the upper half-space, is a circle perpendicular to the bounding plane

The hyperbolic metric is:

This representation of a hyperbolic space is suitable in physics because we have two things. A manifold with finite volume [

Here we present the triangulation of the Seifert spheres and hyperbolic spaces in order to preserve the orientation of manifolds. Those are used in [

Theorem 1. Jaco-Shalen, Johanson Torus Decom-position.

If

This theorem together with the standardization of Thurston’s theorem states: there is a collection of incompressible tori

In the next sections we will do the triangulation of Seifert fibered spaces as homology spheres as well as the hyperbolic space. But first, we see some definitions and constructions that will be helpful.

Definition 3. We consider homology Brieskorn spheres

in a 3-complex dimensional space with the 5-dimensional unit sphere

If

subject to

where

It is possible to represent this relationship by the Euler number of the Seifert fibration of SFH-sphere

which is a topological invariant of

Given a SFH-sphere

and the Seifert invariants are:

where

It is easy to prove that the manifold in (5) with invariants (6) is a homology sphere because its Euler number satisfy (4). We have:

where

Since any SFH-sphere has unique Seifert fibration, the meaning of derivative of the sphere coincides with the derived homology structure of the Seifert fibered space (i.e. acquires characteristics of SFH-spheres).

Definition 4. By induction, we define the derivative

In particular (7) holds for the product of three Seifert invariants

Definition 5. We define here a sequence of SFH-spheres that we call a primary sequence. Let

The primary sequence of the SFH-sphere is defined as:

where

Included (for our model) in this sequence, as its first two terms, ordinary three-dimensional spheres

which are defined as

Recall

and

The mapping

We denote these 3-dimensional spheres (SF-spheres) as

Now we will form the family of manifolds which belongs to the first-primary SFH-spheres and its derivatives:

Definition 6. Using (5) we define the derivative

where

Example: As an example of such derived sequence we present the sequence of the Poincaré homology sphere

Definition 7. Any smooth compact 3-dimensional manifold

The reader can see [

We first take

In

We start our construction with a singular orbit

Regular orbits are represented by

where

To obtain tetrahedra from previously made triangulation we build on each one consisting of two adjacent triangles, three tetrahedra. Taking the first two rectangles on the rectangle 1,2,4,3 we do the following tetrahedra with vertices 3,2,4,7, 1,2,7,3 and 1,7,3,8.

The face (3,4,7,8) of the obtained triangular prism

Therefore the number of tetrahedra to triangular a toroidal neighbourhood of a singular orbit is:

Now, consider a regular orbit

As we can see in

In order to make the complete triangulation including both types of orbits we build on each rectangle consisting of two adjacent triangles, three tetrahedra. See

To have a central orbit we identify the line segment

First we need to construct a cusp hyperbolic space with the following process.

Take the 3-dimensional hyperbolic space

and

We now follow the Thurston geometrization conjecture [

We cut a manifold along essential spheres and tori. We take a connected component of the result of this process, completing spheres with balls and adding the “torus at infinity” in order to obtain the so-called “end” of the manifold. A graphical representation of this is the

Note: If we cut the cusp, we can visualize it as if we cut along a Euclidean plane in the upper half space model and see the previous movements associated to the toroidal boundary. This gives the eigenvalues

Here we construct our hyperbolic space with a single cusp and toroidal boundary.

Consider first the cube

Gluing face

The above construction gives the definition of a cusp.

We need the two-dimensional torus, since in the next section we need to glue the boundary homeomorphically to a torus with either SF-sphere or SFH-sphere.

We describe in this part a triangulation of a cusp hyperbolic space

Based on [

Take a partition from a toroidal boundary, with the number of rectangles

The details are as follows:

We have first

Take the rectangle with vertices

As we need to generalize this process for

For the cube

Note that by gluing

Generalizing this process for any

That is, in

With this we triangulated our cusp hyperbolic space obtaining also the number of generated tetrahedra, given by

In Section 5 we will construct our model and we will use splice diagrams which are similar to graphs.

In our topological model, each fundamental interaction is characterized by the following pair of parameters

Each

We begin by describing the assemblies

We take a Seifert fibered sphere (SF-sphere),

which contains two regular and two singular orbits as described in the Subsection 4.2 and we associate the fol-

lowing splice diagram.

Notation: the second subscript denotes the level

Removing the torus neighbourhoods of singular orbits and introducing the torus part of a cusp hyperbolic space by a splice operation (see [

and the corresponding splice diagram.

With this process we removed singular orbits and exchanged them by cusp hyperbolic spaces.

Now, consider a SF-sphere

In (18) we have glued two hyperbolic spaces, in order to glue in

For the construction of the assembly

In other words, we are indicating that the splice operation was made in singular orbit

Then the assembly will be defined by:

The

Generalizing this idea we get.

Finally, all singular orbits are removed and replaced by either another SFH-sphere or by a cusp hyperbolic space, fulfilling our goal of avoiding singular orbits (no singular points). See [

In our construction we remove the tubular neighbourhoods of singular orbits and replace them by a hyperbolic space thus achieving two things. First, avoiding singularities from any space is closer to a physical representation. Usually it is hard to have a physical representation when in theory we have a singularity. Instead, we can imagine singular points as points at infinity. Second, we construct a different space than Efremov and Mitskievich did, introducing Thurston’s theory and using the work done by Knesser, Milnor, Jaco, Shalen, Johanson. This space shows an improvement for the calculation of the hierarchy of the coupling constants, i.e. reproducing more closely the exponents of the coupling constants. This is important because the use of hyperbolic spaces instead the local use of hyperbolic geometry make a better approach.

Building the model of the universe with 3-dimensional spatial sections from Seifert fibered homology spheres and cusp hyperbolic spaces provides a description of the evolution at least for the hierarchical approach of the coupling constants by means of topological changes. We also use manifolds which contain a finite volume. Therefore, we can calculate volumes in these spaces. In other words, we have an infinite space with finite volume.

We have created here a new cusp hyperbolic space with fibers. This allows calculating the volume of this particular manifold. Once we obtain how to calculate the volume of each tetrahedra in the space, only what we need is to multiply for the number of tetrahedra contained in the cusp hyperbolic space. All of this is due to the space which was discretized. And this could be used in cosmology theories for calculating the volume density of the space.

As a final comment, we emphasize the importance of the approach presented here for the study of alternative cosmological models. Highlighting an important example, the standard Friedmann Singularity-Free (FSF) theories usually consider more complex structures than the simple spherically symmetric point like a Friedmann singularity (the big bang seed with null dimension and infinite density). Following this path, we point out here a possible application of the topological construction introduced in this paper to cosmological scenarios known as “Cusp Cosmology” which are based on the cusp-like geometries resulting from non-linear wave theory [

Thanks to: Ignacio Barradas, Klaus-Peter Schröder, César Caretta, Hugo Garca, Stephanie Dunbar and Lawrence Nash. CONACyT, LAC-INPE, CAPES and PROAP.

Here we take the definition given by [

Definition 8. An orbifold

and an embedding

equivariant with respect to

Next we describe some helpful orbifold characteristics.

Definition 9. A point

Notation: We will denoted by

Note that the pair

Like homology spheres are a particular case of SFM, here is the follow definition.

Definition 10. Homology spheres. Let

with

This manifold is denoted by

By

For any oriented Seifert fibered homology sphere

Let

1) If

then, exists a fibration

2) If

then

We remark that (0.20) determines