TGD diary: Intersection form for 4-manifolds, knots and 2-knots, smooth exotics, and TGD

Thursday, November 10, 2022

Intersection form for 4-manifolds, knots and 2-knots, smooth exotics, and TGD

Gary Ehlenberger sent a highly interesting commentary related to smooth structures in R⁴ discussed in the article of Gompf (see this) and more generally to exotics smoothness discussed from the point of view of mathematical physics in the book of Asselman-Maluga and Brans (see this). I am grateful for these links for Gary. The intersection form of 4-manifold (see this) characterizing partially its 2-homology is a central notion.

The role of intersection forms in TGD

I am not a topologist but I had two good reasons to get interested on intersection forms.

In the TGD framework (see this), the intersection form describes the intersections of string world sheets and partonic 2-surfaces and therefore is of direct physical interest (see this and this).
Knots have an important role in TGD. The 1-homology of the knot complement characterizes the knot. Time evolution defines a knot cobordism as a 2-surface consisting of knotted string world sheets and partonic 2-surfaces. A natural guess is that the 2-homology for the 4-D complement of this cobordism characterizes the knot cobordism. Also 2-knots are possible in 4-D space-time and a natural guess is that knot cobordism defines a 2-knot.
The intersection form for the complement for cobordism as a way to classify these two-knots is therefore highly interesting in the TGD framework. One can also ask what the counterpart for the opening of a 1-knot by repeatedly modifying the knot diagram could mean in the case of 2-knots and what its physical meaning could be in the TGD Universe. Could this opening or more general knot-cobordism of 2-knot take place in zero energy ontology (ZEO) (see this, this and this) as a sequence of discrete quantum jumps leading from the initial 2-knot to the final one.

Why exotic smooth structures are not possible in TGD?

The article of Gabor Etesi (see this) gives a good idea about the physical significance of the existence of exotic smooth structures (see the book and the article). They mean a mathematical catastrophe for both classical relativity and for the quantization of general relativity based on path integral formulation.

The first naive guess was that the exotic smooth structures are not possible in TGD but it turned out that this is not trivially true. The reason is that the smooth structure of the space-time surface is not induced from that of H unlike topology. One could induce smooth structure by assuming it given for the space-time surface so that exotics would be possible. This would however bring an ad hoc element to TGD. This raises the question of how it is induced.

This led to the idea of a holography of smoothness, which means that the smooth structure at the boundary of the manifold determines the smooth structure in the interior. Suppose that the holography of smoothness holds true. In ZEO, space-time surfaces indeed have 3-D ends with a unique smooth structure at the light-like boundaries of the causal diamond CD= cd× CP₂ ⊂ H=M⁴× CP₂, where cd is defined in terms of the intersection of future and past directed light-cones of M⁴. One could say that the absence of exotics implies that D=4 is the maximal dimension of space-time.
The differentiable structure for X⁴⊂ M⁸, obtained by the smooth holography, could be induced to X⁴⊂ H by M⁸-H-duality. Second possibility is based on the map of mass shell hyperboloids to light-cone proper time a=constant hyperboloids of H belonging to the space-time surfaces and to a holography applied to these.
There is however an objection against holography of smoothness (see this). In the last section of the article, I develop a counter argument against the objection. It states that the exotic smooth structures reduce to the ordinary one in a complement of a set consisting of arbitrarily small balls so that local defects are the condensed matter analogy for an exotic smooth structure.