Can one deform an immersion of a 3-manifold in $\mathbb R^4$ to an embedding in $\mathbb R^6$? Let $M^3$ be an oriented 3-manifold, and let $f:M^3\looparrowright \mathbb R^4$ be a codimension one immersion. Is it possible to find a small deformation of the composite map
$$
M^3 \to \mathbb R^4 \to \mathbb R^6
$$
which is an embedding?
(I expect the answer to be "no", and so I'm mostly interested in the method of proof.)
 A: The existence of such an immersion is related to the existence of odd Hopf invariant in $\pi_7(S^4)$. Namely take any such element, for example the homotopy class of the Hopf map. Represent it  - by the Pontrjagin construction - as an embedded framed 3-dimensional submanifold in $R^7$. Using the so called Compression theorem (by Rourke Sanderson, it is more or less equivalent to Smale Hirsch immersion theory) you may isotope this framed immersion so, that one of the framing normal vectors becomes everywhere parallel to the 7-th coordinate direction of $R^7$. Then projecting it to $R^6$ we obtain a framed immersion in $R^6$. The (algebraic) number of double points (they have signs, since the double branches have an ordering, which was higher, which was lower) agrees with the Hopf invariant of the original map. Applying two more times the compression, you can push the immersion into $R^4$ as an immersion. When you lift back from $R^4$ to $R^6$ after arbitrary regular homotopy, the pairity of the double points remains unchanged. (This is the stable Hopf invariant of the element of the stable homotopy class in $\pi^s(3)$ represented by the obtained immersion.)
A: Quoting Theorem F of this paper by Ulrich Koschorke:

For any self-transverse immersion $j$ of a closed 3-manifold $M$ into $\mathbb{R}^4$ the following integers are equal modulo 2:
  
  
*
  
*the Euler number of the surface of double points of $j$;
  
*the number of quadruple points of $j$;
  
*the number of double points of any self-transverse immersion $M\looparrowright\mathbb{R}^6$ which is regularly homotopic to $M \stackrel{j}{\looparrowright}\mathbb{R}^4\subseteq\mathbb{R}^6$.
  
  
  Moreover, there exists an oriented 3-manifold immersed into $\mathbb{R}^4$ such that all
  these numbers are odd. 

An explicit geometric construction of the required immersion is given in the final section.
