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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
T. Deguchi, K. Tsurusaki, A statistical study of random knotting using the Vassiliev invariants. JKTR 3, 321–353 (1994)
T. Kaluza, Zum Unitatsproblem in der Physik, Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.) (1921), pp. 966-972
A. Kawauchi, On quadratic forms of 3-manifolds. Invent. Math. 43, 177–198 (1977)
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
A. Kawauchi, On the signature invariants of infinite cyclic coverings of even dimensional manifolds. Adv. Stud. Pure Math. 9, 177–188 (1986)
A. Kawauchi, The imbedding problem of 3-manifolds into 4-manifolds. Osaka J. Math. 25, 171–183 (1988)
A. Kawauchi, Knots in the Stable 4-Space; An Overview, A Fete of Topology (Academic, Cambridge, 1988), pp. 453–470
A. Kawauchi, On the fundamental class of an infinite cyclic covering. Kobe J. Math. 15, 103–114 (1998)
A. Kawauchi, The quadratic form of a link, in Proceedings of the Low Dimension Topology. Contemporary Mathematics, vol. 233 (1999), pp. 97–116
A. Kawauchi, Component-conservative invertibility of links and Samsara 4-manifolds on 3-manifolds. Asia Pac. J. Math. 1, 86–106 (2014)
A. Kawauchi, A Survey of Knot Theory (Birkhäuser, 1996)
A. Kawauchi, On 4-dimensional universe for every 3-dimensional manifold. Topol. Appl. 196, 575–593 (2015)
A. Kawauchi, Characteristic genera of closed orientable 3-manifolds. Kyungpook Math. J. 55, 753–771 (2015)
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)
A. Kawauchi, S. Kojima, Algebraic classification of linking pairings on 3-manifolds. Math. Ann. 253, 29–42 (1980)
A. Kawauchi, I. Tayama, Representing 3-manifolds in the complex number plane. Topol. Appl. 230C, 425–443 (2017)
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
O. Klein, Quantentheorie und funfdimensionale Relativitatstheorie. Zeitschrift fur Physik A. 37(12), 895–906 (1926)
T. Matumoto, Extension problem of diffeomorphisms of a 3-torus over some 4-manifolds. Hiroshima Math. J. 14, 189–201 (1984)
V.S. Netchitailo, Mathematical overview of hypersphere world-universe model. J. High Energy Phys. Gravit. Cosmol. 3, 415–437 (2017)
L. Randall, R. Sundrum, Large mass hierarchy from a small extra dimension. Phys. Rev. Lett. 83, 33700 (1999). hep-ph/9905221
L. Randall, R. Sundrum, An alternative to compactification. Phys. Rev. Lett. 83, 4690 (1999). hep-th/9906064
T. Shiomi, On imbedding 3-manifolds into 4-manifolds. Osaka J. Math. 28, 649–661 (1991)
E. Uehara, T. Deguchi, Characteristic length of the knotting probability revisited, J. Phys.: Condens. Matter 27, 354104 (9pp) (2015)
E. Uehara, T. Deguchi, Knotting probability of self-avoiding polygons under a topological constraint. J. Chem. Phys. 147, 094901–16 (2017)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-981-13-5742-8_9
Published:
Publisher Name: Birkhäuser, Singapore
Print ISBN: 978-981-13-5741-1
Online ISBN: 978-981-13-5742-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)