Homological Infinity of 4D Universe for Every 3-Manifold

  • Akio KawauchiEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)


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.


4D universe 4D punctured universe Topological index Collision modification 3-manifold Punctured 3-manifold Signature theorem 

Mathematics Subject Classification (2010)

Primary: 57N13 Secondary: 57M27 57N35 


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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Osaka City University Advanced Mathematical Institute SugimotoSumiysoshi-ku, OsakaJapan

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