Hi Joy,
In our context, that's a cool picture in the attachment! Instead of spaces being split off in the way we have been suggesting, the diagram gives a visual image of S15 exploding apart.
The inner lines give the physical spaces of my model, where the homotopy group for the map from S7 to S3 gives the chiral Z2 of the Higgs vacuum. The space S2 is that enclosing a monopole and the homotopy group for the map from S6 to S2 gives Z12=Z3*Z4 for a 3 by 4 table of topological monopoles S0 (i.e. the correct particles). The homotopy groups for maps from S6 to S4 and S4 to S2 are both Z2, suggesting compatibility with the Z2 of the Higgs vacuum (S7->S3).
For physics, there would seem to be a difference between the spaces of the inner lines and those of the outer lines. If we look at all the normed division algebras, R, C, H, O the total dimension is 15, but R is occurring here 4 times: once on its own and once in each of C, H and O. My model results if we say that there can't really be 4 separate occurrences of R, so subtract the over counting of R to give the dimension of the combined R, C, H, O as being 12 - where the algebras coincide on R. With the meta-principle of "make no preference" and a closure condition this gives my space of S0, S1, S3, S7.
On the other hand, the S2 is a monopole enclosing space, S6 is the space left after splitting off the unbroken S1 of electromagnetism - which is responsible for the topological monopoles - from S7, and S4 is your hidden domain. It is noticeable that all these spaces are of a different character from the "physical spaces" S1, S3, S7.
A critical thing to explain here is still why is time different? If time were associated with R in some way, and the S3 and S7 spaces with the imaginary parts of H and O we could perhaps get a natural looking space-time split.
Best,
Michael
