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Homological Infinity of 4D Universe for Every 3-Manifold

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Algebraic Topology and Related Topics

Part of the book series: Trends in Mathematics ((TM))

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Abstract

This chapter is an explanation on recent investigations on homological infinity of a 4D universe for every 3-manifold, namely a boundary-less connected oriented 4-manifold with every closed connected oriented 3-manifold embedded, and homological infinity of a 4D punctured universe, namely a boundary-less connected oriented 4-manifold with every punctured 3-manifold embedded. Types 1, 2, and full 4D universes are introduced as fine notions of a 4D universe. After introducing some topological indexes for every (possibly non-compact) oriented 4-manifold, we show the infinity on the topological indexes of every 4D universe and every 4D punctured universe. Further, it is observed that a full 4D universe is produced by collision modifications between 3-sphere fibers in the 4D spherical shell (i.e., the 3-sphere bundle over the real line) embedded properly in any 5-dimensional open manifold and the second rational homology groups of every 4D universe and every 4D punctured universe are always infinitely generated over the rationals.

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References

  1. T. Deguchi, K. Tsurusaki, A statistical study of random knotting using the Vassiliev invariants. JKTR 3, 321–353 (1994)

    MathSciNet  MATH  Google Scholar 

  2. T. Kaluza, Zum Unitatsproblem in der Physik, Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.) (1921), pp. 966-972

    Google Scholar 

  3. A. Kawauchi, On quadratic forms of 3-manifolds. Invent. Math. 43, 177–198 (1977)

    Article  MathSciNet  Google Scholar 

  4. A. Kawauchi, On The signature invariants of infinite cyclic coverings of closed odd dimensional manifolds, in Algebraic and Topological Theories-to the Memory of Dr. T. Miyata (Kinokuniya Co. Ltd., 1985), 52–85

    Google Scholar 

  5. A. Kawauchi, On the signature invariants of infinite cyclic coverings of even dimensional manifolds. Adv. Stud. Pure Math. 9, 177–188 (1986)

    MathSciNet  Google Scholar 

  6. A. Kawauchi, The imbedding problem of 3-manifolds into 4-manifolds. Osaka J. Math. 25, 171–183 (1988)

    MathSciNet  MATH  Google Scholar 

  7. A. Kawauchi, Knots in the Stable 4-Space; An Overview, A Fete of Topology (Academic, Cambridge, 1988), pp. 453–470

    MATH  Google Scholar 

  8. A. Kawauchi, On the fundamental class of an infinite cyclic covering. Kobe J. Math. 15, 103–114 (1998)

    MathSciNet  MATH  Google Scholar 

  9. A. Kawauchi, The quadratic form of a link, in Proceedings of the Low Dimension Topology. Contemporary Mathematics, vol. 233 (1999), pp. 97–116

    Google Scholar 

  10. A. Kawauchi, Component-conservative invertibility of links and Samsara 4-manifolds on 3-manifolds. Asia Pac. J. Math. 1, 86–106 (2014)

    MATH  Google Scholar 

  11. A. Kawauchi, A Survey of Knot Theory (Birkhäuser, 1996)

    Google Scholar 

  12. A. Kawauchi, On 4-dimensional universe for every 3-dimensional manifold. Topol. Appl. 196, 575–593 (2015)

    Article  MathSciNet  Google Scholar 

  13. A. Kawauchi, Characteristic genera of closed orientable 3-manifolds. Kyungpook Math. J. 55, 753–771 (2015)

    Article  MathSciNet  Google Scholar 

  14. A. Kawauchi, Topology of 4D universe for every 3-manifold, in Proceedings of the second Pan-Pacific International Conference on Topology and Applications. Topol. Appl. (to appear)

    Google Scholar 

  15. A. Kawauchi, S. Kojima, Algebraic classification of linking pairings on 3-manifolds. Math. Ann. 253, 29–42 (1980)

    Article  MathSciNet  Google Scholar 

  16. A. Kawauchi, I. Tayama, Representing 3-manifolds in the complex number plane. Topol. Appl. 230C, 425–443 (2017)

    Article  MathSciNet  Google Scholar 

  17. J. Khoury, B.A. Ovrut, P.J. Steinhardt, N. Turok, Ekpyrotic universe: colliding branes and the origin of the hot big bang. Phys. Rev. D 64, 123522 (2001). hep-th/0103239

    Article  MathSciNet  Google Scholar 

  18. O. Klein, Quantentheorie und funfdimensionale Relativitatstheorie. Zeitschrift fur Physik A. 37(12), 895–906 (1926)

    Article  Google Scholar 

  19. T. Matumoto, Extension problem of diffeomorphisms of a 3-torus over some 4-manifolds. Hiroshima Math. J. 14, 189–201 (1984)

    Article  MathSciNet  Google Scholar 

  20. V.S. Netchitailo, Mathematical overview of hypersphere world-universe model. J. High Energy Phys. Gravit. Cosmol. 3, 415–437 (2017)

    Article  Google Scholar 

  21. L. Randall, R. Sundrum, Large mass hierarchy from a small extra dimension. Phys. Rev. Lett. 83, 33700 (1999). hep-ph/9905221

    MathSciNet  MATH  Google Scholar 

  22. L. Randall, R. Sundrum, An alternative to compactification. Phys. Rev. Lett. 83, 4690 (1999). hep-th/9906064

    Article  MathSciNet  Google Scholar 

  23. T. Shiomi, On imbedding 3-manifolds into 4-manifolds. Osaka J. Math. 28, 649–661 (1991)

    MathSciNet  MATH  Google Scholar 

  24. E. Uehara, T. Deguchi, Characteristic length of the knotting probability revisited, J. Phys.: Condens. Matter 27, 354104 (9pp) (2015)

    Google Scholar 

  25. E. Uehara, T. Deguchi, Knotting probability of self-avoiding polygons under a topological constraint. J. Chem. Phys. 147, 094901–16 (2017)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Osaka City University Advanced Mathematical Institute Sugimoto, Sumiysoshi-ku, Osaka, 558-8585, Japan

    Akio Kawauchi

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  1. Akio Kawauchi
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Correspondence to Akio Kawauchi .

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Editors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, SAS Nagar, Punjab, India

    Mahender Singh

  2. Department of Mathematics, Inha University, Incheon, Korea (Republic of)

    Yongjin Song

  3. Department of Mathematics, National University of Singapore, Singapore, Singapore

    Jie Wu

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Kawauchi, A. (2019). Homological Infinity of 4D Universe for Every 3-Manifold. In: Singh, M., Song, Y., Wu, J. (eds) Algebraic Topology and Related Topics. Trends in Mathematics. Birkhäuser, Singapore. https://doi.org/10.1007/978-981-13-5742-8_9

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