On 4-dimensional universe for every 3-dimensional manifold

A boundary-less connected oriented 4-manifold is called a universe for every 3-manifold if every closed connected oriented 3-manifold is embedded in it, and a punctured universe if every punctured 3-manifold is embedded in it, which is known to be an open 4-manifold. We introduce types 1, 2 and full universes as refined notions of a universe and a punctured universe and investigate some relationships among them. After introducing some topological invariants for every (possibly non-compact) oriented 4-manifold which we call the topological indexes, we show infinity and independence on some topological indexes of every universe.