Archive for January, 2017

Of course I hold the firm conviction that not only is our actual universe a compact three dimensional hypersurface of a fixed four-sphere of fixed radius (which is responsible for all quantum phenomena) but having been convinced after some years that unless I connect up my views with the mainstream this will not be accepted, I am slowly working on studying standard material.  In fact the geometry of the situation I am fairly sure is true in actual reality can be considered through the lens of topological quantum field theories, where one considers a three-manifold Y that is a submanifold of a four-manifold X.  Dijkgraaf and Witten had a nice paper for which they consider in what way the Chern-Simons action on a three manifold is related to the Yang-Mills action on the bounding four-manifold.  This is an interesting thing to do for our situation.  A useful analysis of theirs is the following:  if you take a principal G-bundle on Y and want to extend this bundle to X the four-manifold, this can be accomplished for a compact Lie group case essentially because the odd dimensional homology groups H_{k+1}(BG,\mathbf{Z})=H_{k}(G,\mathbf{Z})=0.  So you get this from the fact that the short exact sequence G \rightarrow EG \rightarrow BG has a contractible space in the middle (by definition of classifying space) and then taking the long exact sequence in homology to get an isomorphism of the homology of BG and $G$ in the appropriate indices, then use Poincare duality and the fact that the cohomology of compact simple groups are known — Milnor’s Morse Theory has a proof that the loop space over G not only has the homotopy type of a CW complex but has only even dimensional skeleta and then there is an isomorphism of loop group cohomology and that of the Lie group.  The conclusion is that $H_3(BG,\mathbf{Z})=0$ for compact Lie groups such as SU(n).   The Witten Dijkgraaf (dijkgraaf-witten) procedure is to consider the classifying map associated to a bundle E\rightarrow Y, say \gamma:Y\rightarrow BG and consider whether the image is the boundary of some four-manifold in BG.  The image \gamma_*[Y] has no boundary and is three-dimensional therefore is represented in $H_3(BG)$ and since this is trivial for compact Lie groups of interest we can extend our classifying map over X and then pull back the universal bundle to X.  This is a very nice method.  Now in our case we have an ambient four-sphere and we actually know the instantons over the four-sphere by the wonderful results of Atiyah-Drinfeld-Hitchin-Manin who produced explicit solutions for instantons for a family of principal bundles on S^4.

Gromov came up with a very interesting set of concepts such as ‘filling radius’ and ‘filling volume’ — see here (filling-riemannian-manifolds.  Filling radius is essentially the radius of the largest ball that fits inside the manifold if it is isometrically embedded and filling volume is defined similarly.  Interesting bounds involving these are FillRad(Y)^n \le FillVol(Y) \le C_n Vol(Y)^{(n+1)/n} and FillRad(M) \ge InjRad(M)/2(dim M + 2).  So we can obtain an upper bound on the injectivity radius of our universe and a bound for the volume as well by following geometric analysis of this type.




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