AM  Vol.5 No.6 , April 2014
An Alternative Manifold for Cosmology Using Seifert Fibered and Hyperbolic Spaces
ABSTRACT

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.


Cite this paper
Mejía, M. and Rosa, R. (2014) An Alternative Manifold for Cosmology Using Seifert Fibered and Hyperbolic Spaces. Applied Mathematics, 5, 1013-1028. doi: 10.4236/am.2014.56096.
References
[1]   Hawking, S. (1988) Wormholes in Spacetime. Physical Review D, 37, 904.
http://dx.doi.org/10.1103/PhysRevD.37.904

[2]   Weinberg, S. (1989) The Cosmological Constant Problem. Reviews of Modern Physics, 61, 1.
http://dx.doi.org/10.1103/RevModPhys.61.1

[3]   Horowitz, G.T. (1991) Topology Change in Classical and Quantum Gravity. Classical and Quantum Gravity, 8, 587-601.
http://dx.doi.org/10.1088/0264-9381/8/4/007

[4]   De Lorenci, V.A., Martin, J., Pinto-Neto, N. and Soares, I.D. (1997) Topology Change in Canonical Quantum Cosmology.

[5]   Kronheimer, P.B. and Mrowka, T.S. (1993) Gauge Theory for Embedded Surfaces. Topology, 32, 773-826.
http://dx.doi.org/10.1016/0040-9383(93)90051-V

[6]   Efremov, V.N. (1996) Flat Connection Contribution to Topology Changing Amplitudes in an Ensemble of Seifert fibered Homology Spheres. International Journal of Theoretical Physics, 35, 63-68.
http://dx.doi.org/10.1007/BF02082934

[7]   Efremov, V.N. (1997) Siebenman-Type Cobordisms with Borders and Topology Changes by Quantum Tunnelling. International Journal of Theoretical Physics, 36, 1133-1151.
http://dx.doi.org/10.1007/BF02435804

[8]   Efremov, V.N. and Mitskievitch, N.V. (2003) Discrete Model of Space-Time in Terms of Inverse Spectra of the T0 Alexandroff Topological Spaces.

[9]   Efremov, V.N. and Mitskievitch, N.V. (2004) T Discrete Model of Space-Time in Terms of Inverse Spectra of the T0 Alexandroff Topological Spaces Opology Changes in Terms of Proper Inverse Spectra of T0-Discrete Spaces and Hierarchy of Fundamental Interactions in a Universe Glued Together of Seifert Homologic Spheres. In: Progress in General Relativity and Quantum Cosmology Research, Nova Sci. Publishers.

[10]   Mejia, M.E. (2009) Un Modelo del Universo y la Comparación de la Jerarquia de las Constantes de Acoplamiento Experimentales con las Constantes de Acoplamiento del Modelo. Memoria de los extensos del VI Encuentro Participación de la Mujer en la Ciencia, 1, 46-47.

[11]   Barrett, J.W. (1995) Quantum Gravity as Topological Quantum Field Theory. Journal of Mathematical Physics, 36, 6161.
http://dx.doi.org/10.1063/1.531239

[12]   Carlip, S. (2005) Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe. Living Reviews in Relativity, 8, 1.

[13]   Anderson, M., Carlip, S., Ratcliffe, J.G., Surya, S. and Tschantz, S.T. (2004) Peaks in the Hartle-Hawking Wave Function from Sums over Topologies Class. Quantum Gravity, 21, 729.
http://dx.doi.org/10.1088/0264-9381/21/2/025

[14]   Bytsenko, A.A. and Guimaraes, M.E.X. (2008) Expository Remarks on Three-Dimensional Gravity and Hyperbolic Invariants.

[15]   Kneser, H. (1929) Jahresbericht der Deutshen. Mathematiker-Vereinigung, 38, 248-260.

[16]   Milnor, J. (1962) A Unique Decomposition Theorem for 3-Manifolds. American Journal of Mathematics, 84, 1-7.

[17]   Eisenbud, D. and Newmann, W.D. (1985) Three-Dimensional Link Theory and Invariants of Plane Curve Singularities. Ann. Math. Stud. Princeton University Press.

[18]   Johanson, K. (1979) Homotopy Equivalences of 3-Manifolds with Boundaries. Lecture Notes in Math. N 1-761. Springer-Verlag, Berlin.

[19]   Jaco, W. and Shalen, P.B. (1979) Seifert Fibered Spaces in 3-Manifolds. American Mathematical Society, 220.

[20]   Thurston W. P. (1982) Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. American Mathematical Society. Bulletin. New Series, 6, 357-381.

[21]   Thurston, W.P. (1997) The Geometry and Topology of 3-Manifolds. Electronic Version 1.0.
http://library.msri.org/books/gt3m/

[22]   Thurston, W. P. (1997) Three-Dimensional Geometry and Topology. Vol. 1, Princeton University Press, Princeton.

[23]   Jaco, W. (1980) Lectures on 3-Manifold Topology. Regional Conference Series in Mathematics. American Mathematical Soc.

[24]   Newmann, W. D. and Zagier, D. (1985) Volumes of Hyperbolic Three-Manifolds. Topology, 24, 307-332.
http://dx.doi.org/10.1016/0040-9383(85)90004-7

[25]   Aschenbrenner, M. Friedl, S. and Wilton, H. (2013) 3-Manifold Groups. e-print arXiv:1205.0202 [math.GT].

[26]   Rong,Y. (1992) Degree One Maps between Geometric 3-Manifolds. American Mathematical Society, 332, 411-436.

[27]   Hernándes, A.M. (2003) Cambios Topológicos y Jerarqua de las Interacciones Fundamentales en un Modelo Cosmológico T0-Discreto del Universo Construido de Esferas Homológicas Fibradas de Seifert. Tesis para obtener el grado de maestro en ciencias en la Universidad de Guadalajara.

[28]   Milnor, J. (1975) On the 3-Dimensional Brieskorn Manifolds M (p,q,r).
http://www.maths.ed.ac.uk/126aar/papersmilnbries.pdf

[29]   Rosa, R.R., Strieder, C. and Stalder. D.H. (2012) An Alternative Singularity-Free Cosmological Scenario from Cusp Geometries. AIP Conference Proceedings, 1483, 441-446.
http://dx.doi.org/10.1063/1.4756991

[30]   Milnor, J. (1958) Review: Norman Steenrod, The Topology of Fibre Bundles. Bulletin of the American Mathematical Society, 64, 202-203. http://dx.doi.org/10.1090/S0002-9904-1958-10211-6

 
 
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