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.)



Can one deform an immersion of a 3-manifold in $\mathbb R^4$ to an embedding in $\mathbb R^6$?
Quoting Theorem F of [this paper](http://www.maths.ed.ac.uk/~aar/papers/koschork.pdf) 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.


