**I have already asked this question on MSE https://math.stackexchange.com/questions/4272279/1-dimensional-foliation-of-surfaces-with-prescribed-graph-of-foliation **
Definition of the graph of a foliation
Let we have a $k$ dimensional foliation of an $n$ dimensional manifol $M$. One associates to this foliated manifold a (not necessarily Hausdorff) n+k dimensional manifold called foliation groupoid:
$G=\{(x,y,\alpha)|\; x,y\quad \text{lie on the same leaf and}\; \alpha \;\text{is a curve on the leaf which join}\; x \;to \; y\}$
We equip $G$ with an equivalent relation: $(x,y,\alpha) \sim (x,y,\beta)$ if the holonomy associated to $\alpha\circ \beta ^{-1}$ is the identity. We denote $G/\sim$ again by $G$
It is a manifold of dimension $n+k$ it is Hausdorff if the foliation is real analytic. $M$ is considered as a subset of $G$ in an obvious manner: $M\ni x\mapsto (x,x,\alpha_x)$ where $\alpha_x$ is the constant curve. It is groupoid with obvious maps $r,s:G\to M$ with $r(x,y,\alpha)=y, s(x,y,\alpha) =x$. The foliation charts of $M$ (and transversal sections) gives us the manifold charts for $G$.
According to the above definition of the graph of a foliation by Winkelnkemper we ask the following questions:
Let $G$ be one of the following non Hausdorff 3 dim manifolds
- $G$ is a 3 sphere with two north poles.That is: We consider the disjoint union of two $S^{3}$. Then we identify each point $x$ of $S^{3}\setminus \{N\}$ to itself in the other copy. More precisely the equivalent relation is defined as follows:
A 3-sphere with two Northpole is the quotient of the following equivalent relation on space $X$ below:
$X=S^3\times\{0\} \bigcup S^3\times \{1\}$.
For $x \neq N$ we define $(x,0)\sim (x,1)$. The quotìent space is called a 3-sphere with two north pole
2)$G$ is a 3 sphere with two equators
3)$G$ is an space($\mathbb{R}^{3}$) with two origins
Is there a one dimensional foliation of a surface whose graph of the foliation is diffeomorphic to $G$?
If the answer is no, we continue as follows:
Is there an interesting smooth groupoid structure on $G$? And what is the structure of the corresponding $C^{*}$ algebra of the space of continuous sections of half densitities?(What type of explicit $C^{*}$ algebras would appear?)
