The \Gamma-sets and \zeta-space.

Assume X is a smooth convex open manifold and \partial X is a convex regular polytope.

Def: An *\mathcal F-completion,* or *foliational completion* of a *block*, \mathcal B:=X\cup\partial X over its vertex set V, is \mathscr CX_V:=\cup_{v_{ij}\in V} \mathcal F_{v_{ij}} where i,j index the vertices. 

Example: Set X=(0,1)^n. Take a smooth foliation \mathcal F for every pair of points v_i,v_j\in \partial X=[0,1]^n-(0,1)^n satisfying \mathrm{sup~dist}_n(v_i,v_j)=\sqrt{r}. Let the leaves in each \mathcal F be mutually diffeomorphic to the class M=(0,\sqrt{n})\times S^{n-2} accumulating to v_i,v_j. We see that r=\lbrace 1,\sqrt{n} \rbrace. Allowing only r=\sqrt{n}, we can look at sections of \mathscr CX_V:

Def: A \Gamma-set on \mathscr CX_V is the union of all leaf intersections of the foliations i.e. 

\Gamma_{n}:= \lbrace \mathcal (\mathcal F_1 \cap \mathcal F_2)\cup (\mathcal F_2 \cap \mathcal F_3) \cup (\mathcal F_3 \cap \mathcal F_4) \cup \cdot\cdot\cdot \cup (\mathcal F_{n-1} \cap \mathcal F_{n})  \rbrace.

Then it follows trivially that for any given foliation, \Gamma_1=X^1=(0,1) and \Gamma_2=X^2=(0,1)^2. 

>What is \Gamma_3 equal to?

It seems that \Gamma_3 is equal to three mutually orthogonal planes but I want some reassurance that this is true. I don't know the structure of the \Gamma-sets for n>3.

The following is meant to provide context as to why I am defining these concepts above in the first place:

Personally I am interested in the \Gamma-sets because if you define vector fields on elements of the M-class, you can add the vectors along sections of the \Gamma-sets resulting in vector bundles that alternate orientation when traveling along a path on the \Gamma-set. Thus the \Gamma-sets tell you about the geometry.

Another reason I'm interested in the construction in general is because one inherits a notion of decay with \mathscr CX_V. Specifically, the M-class rapidly decays to zero when traveling in the direction prescribed by the vector fields placed on the M-class. This can be formalized analogous to rapid decay of elements of the Schwartz space \mathcal S(\Bbb R^n), which are functions, not surfaces. However it is not difficult to adapt this notion.

With this rapid decay we have a concept of a Schwartz space which plays nice with zeta functions because it is known that we can build zeta functions from elements in the Schwartz space.

   

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