First of all it is a trivial observation that in the S4 model the physical universe must be a hypersurface topologically into just by inverse stereographic projection from any point not in the physical hypersurface and is diffeomorphic to . I just don’t know very much about three-manifold topology so I have to defer to the master of topology such as Michael Freedman on this issue.

Of course I am interested in having an exact sharp criterion from topologists about the obstructions to embeddability to because these are the topological restrictions on the physical universe itself and so this question is exceedingly important, and it is a purely topological question. I am very proud of my discovery that the actual universe is a round four sphere partly because we can ask this question and take the mathematical answer seriously as question about Nature. The fact that a great topologist like Michael Freedman is thinking about the issue is also encouraging. So that the topological restriction from the pure embeddability point of view. On the other hand one could imagine that the topology of the three manifold would not be very different from the equator by soft reasoning that well, there’s not really much matter in the universe compared to its size so how complicated can the topology be? I think this is * deceptive*. My intuition tells me that there is some topological complexity that arises from the complications in the microscopic scales. I am being vague about this still but a different way of seeing the point is that we are not dealing with a simple mean curvature flow type situation but a Navier-Stokes equation in the macroscopic scale and we don’t have much clue about

*of the singularities of a Navier-Stokes equation. One of the reasons I sort of dance around the edges on these issues is because I don’t have a good sense of which point of view might show more of the the fundamental geometry of the universe as an evolving three manifold. Of course I am frustrated that the best minds of our time are wasting their time on string theory and quantum field theory rather than focusing on the real issues of what’s at stake. But this is probably unfair and I don’t have the energy or interest of trying to convince people that this is where they should be focused on: the problem of three dimensional topology of the universe is the same as these two problems: which three manifolds can embed into and what topological complexity can arise from starting at the equator and following the Navier-Stokes nonlinear parabolic equation and understand the topological changes in the resulting hypersurfaces. Sure, Floer Homology type theories might do the job. I obtained a copy of Mrowka and Kronheimer’s Monopoles and Three Manifolds to get a better sense of where this can lead. I am reading a paper of Giga to get a better sense of how to solve the Navier-Stokes equation even though I have an explicit scheme for solving it. I don’t think I will get the right way to see this without some perspective. It’s exciting but frustrating to have the greatest opportunity to be staring at the heart of Nature and not have the right tools handy. I was thinking before of doing something simple like modeling the hypersurface as fluctuating Brownian motion but now that I am looking at this a bit clearer, it’s better not to waste my time on artificial models and get the right model at first shot. I’m not getting any younger.*

**topological complexity**