Division-Algebras/Poincare-Conjecture Correspondence

Abstract

We briefly describe the importance of division algebras and Poincaré conjecture in both mathematical and physical scenarios. Mathematically, we argue that using the torsion concept one can combine the formalisms of division algebras and Poincaré conjecture. Physically, we show that both formalisms may be the underlying mathematical tools in special relativity and cosmology. Moreover, we explore the possibility that by using the concept of n-qubit system, such conjecture may allow generalization the Hopf maps.

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J. Nieto, "Division-Algebras/Poincare-Conjecture Correspondence," Journal of Modern Physics, Vol. 4 No. 8A, 2013, pp. 32-36. doi: 10.4236/jmp.2013.48A005.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] R. Bott and J. Milnor, Bulletin of the American Mathematical Society, Vol. 64, 1958, pp. 87-89. doi:10.1090/S0002-9904-1958-10166-4
[2] M. A. Kervaire, Proceedings of the National Academy of Sciences, Vol. 44, 1958, p. 280.
[3] J. Milnor, Annals of Mathematics, Vol. 68, 1958, p. 444. doi:10.2307/1970255
[4] J. F. Adams, Annals of Mathematics, Vol. 75, 1962, p. 603. doi:10.2307/1970213
[5] J. A. Nieto and L. N. Alejo-Armenta, International Journal of Modern Physics A, Vol. 16, 2001, p. 4207. doi:10.1142/S0217751X01005213
[6] M. F. Atiyah, “K-Theory,” Benjamin, Inc., New York, Amsterdam, 1967.
[7] C. Nash and S. Sen, “Topology and Geometry for Physicists,” Academic Press, Cambridge, 1983.
[8] I. L. Kantor and A. S. Solodovnikov, “Hypercomplex Numbers—An Elementary Introduction to Algebras,” Springer, Berlin, 1989.
[9] F. Hirzebruch, “Numbers,” In: S. Axier, F. W. Gehring and P. R. Halmos, Eds., Numbers, Springer, New York, 1991, p. 281.
[10] J. Baez, Bulletin of the American Mathematical Society, Vol. 39, 2002, pp. 145-205. doi:10.1090/S0273-0979-01-00934-X
[11] B. Kleiner and J. Lott, Geometry and Topology, Vol. 12, 2008, pp. 2587-2855.
[12] J. W. Morgan and G. Tian, “Ricci Flow and the Poincare Conjecture,” Clay Mathematics Monographs, Vol. 3, American Mathematical Society, Providence, 2007.
[13] H.-D. Cao and X.-P. Zhu, Asian Journal of Mathematics, Vol. 10, 2006, p. 165.
[14] R. Hamilton and J. Diff, Journal of Differential Geometry, Vol. 17, 1982, pp. 255-306.
[15] L. Borsten, M. J. Duff and P. Levay, Classical and Quantum Gravity, Vol. 29, 2012, Article ID: 224008. doi:10.1088/0264-9381/29/22/224008
[16] M. Green, J. Schwarz and E. Witten, “Superstring Theory,” Cambridge University Press, Cambridge, 1987.
[17] T. Kugo and P. Townsend, Nuclear Physics B, Vol. 221, 1983, pp. 357-380. doi:10.1016/0550-3213(83)90584-9
[18] J. M. Evans, Nuclear Physics B, Vol. 298, 1988, pp. 92-108. doi:10.1016/0550-3213(88)90305-7
[19] R. Mosseri and R. Dandoloff, Journal of Physics A: Mathematical and General, Vol. 34, 2001, Article ID: 10243. doi:10.1088/0305-4470/34/47/324
[20] R. Mosseri, Topology in Condense Matter, Springer Series in Solid State Sciences, Vol. 150, 2006, pp. 187-2003.
[21] B. A. Bernevig and H. D. Chen, Journal of Physics A, Vol. 36, 2003, p. 8325. doi:10.1088/0305-4470/36/30/309
[22] M. D. Maia. International Journal of Modern Physics A, Vol. 26, 2011, pp. 3813-3820.
[23] D. O’Shea, “The Poincare Conjecture: In Search of the Shape of the Universe,” Walker & Company, New York, 2007.
[24] I. Bakas, D. Orlando and P. M. Petropoulos, JHEP, Vol. 0701, 2007, p. 040.
[25] J. Isenberg and M. Jackson, Journal of Differential Geometry, Vol. 35, 1992, p. 723.
[26] F. Bourliot, J. Estes, P. M. Petropoulos and P. Spindel, Physical Review D, Vol. 81, 2010, Article ID: 104001.
[27] R. Dundarer, F. Gursey and C. H. Tze, Journal of Mathematical Physics, Vol. 25, 1984, p. 1496. doi:10.1063/1.526321
[28] M. Ryan and L. Shepley, “Homogeneous Relativistic Cosmologies,” Princeton University Press, Princeton, 1975.
[29] A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, “Oriented Matroids,” Cambridge University Press, Cambridge, 1999.
[30] J. G. Oxley, “Matroid Theory,” Oxford University Press, New York, 2006.
[31] J. A. Nieto, Advances in Theoretical and Mathematical Physics, Vol. 8, 2004, p. 177.
[32] J. A. Nieto, Advances in Theoretical and Mathematical Physics, Vol. 10, 2006, p. 747.
[33] H. Whitney, Transactions of the American Mathematical Society, Vol. 34, 1932, pp. 339-362. doi:10.1090/S0002-9947-1932-1501641-2
[34] H. Whitney, The American Journal of Mathematics, Vol. 57, 1935, p. 509. doi:10.2307/2371182
[35] J. P. S. Kung, “A Source Book on Matroid Theory,” Birkhauser Boston, Inc., Massachusetts, 1986. doi:10.1007/978-1-4684-9199-9

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