**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

1) $G$  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?)