Topology of a 4D universe for every 3-manifold

Abstract

A 4D universe is a 4-dimensional boundary-less connected oriented manifold with every closed 3-manifold (i.e., a 3-dimensional closed connected oriented manifold) embedded. A 4D punctured universe is a 4-dimensional boundary-less connected oriented manifold with the punctured manifold of every closed 3-manifold embedded. Every 4D universe and every 4D punctured universe are open 4-dimensional manifolds. If a closed 3-manifold is considered as a 3D universe, then every 4D spacetime is embedded in every 4D universe and hence every 4D universe is a classifying space for every spacetime. In this paper, 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. As a previous result, it was shown that any 4D universe and 4D punctured universe must have infinity on some homological indexes. It is shown in this paper that the second rational homology groups of every 4D universe and every 4D punctured universe are always infinitely generated.