Johansson's theorem states the following:


Given $f:M\_1\rightarrow M\_2$ (not a pair map) an homotopy equivalence between 3-manifolds with incompressible boundary. 


Let $V\_i$ be the components of the characteristic submanifolds meeting the boundary then we can homotope $f\simeq g$ so that $g$ is an homotopy equivalence on $V\_i$ and is an homeomorphism on $\overline{M\_1\setminus V\_1}\rightarrow \overline{M\_2\setminus V\_2}$.


I also know about Waldhausen's theorem that if $f$ sends $\partial M\_1$ to $\partial M\_2$ homeomorphically then $f$ can be homotoped rel $\partial M\_1$ to a homeomorphism. 


My question is that if we know that $f\vert\_{\partial M\_1}$ is an embedding can we homotope $f:M\_1\rightarrow M\_2$ rel boundary to be an embedding?



On Johansson's Theorem on homotopy equivalences of 3-manifolds
Yes, this is true, with the appropriate assumption that $M\_1$ is irreducible (this is needed for Johansson and Waldhausen's statements), 
and let's say orientable.


One may reduce to Waldhausen's theorem if we know that $f\_{|\partial M\_1}$ is homotopic to a homeomorphism $f'\_{|\partial M\_1}:\partial M\_1\to \partial M\_2$, by homotopy extension. 


Consider the image surface $f(\partial M\_1) \subset M\_2$ (assume this embedding is transverse, so doesn't meet $\partial M\_2$). Since $\partial M\_1$ is homologically trivial in $M\_1$, $f(\partial M\_1)$ is homologically trivial in $M\_2$, hence bounds a submanifold $M \subset M\_2$. Then $\partial M =f(\partial M\_1)$ is homotopic to $\partial M\_2$ iff $M\_2-M$ is a product manifold (this follows from a result of Waldhausen, see Corollary 5.5 of 


*Friedhelm Waldhausen*, MR 224099 [**On irreducible $3$-manifolds which are sufficiently large**](http://www.ams.org/mathscinet-getitem?mr=224099), *Ann. of Math. (2)* **87** (1968), 56--88.). 


Take a homotopy inverse $g:M\_2\to M\_1$ to $f$. Then $g\_{|M}$ is $\pi\_1$-injective, and we may homotope $g$ so that $g\_{|\partial M}\to \partial M\_1$ is a homeomorphism. This map is also degree one, and $\pi\_1$-injective, hence a homotopy equivalence. But this implies that $\pi\_1M\to \pi\_1 M\_2$ is an isomorphism. In turn, this implies that $M\_2-M$ is a product. 


(I'm leaving out many details here, and I don't mean to give a proof by intimidation, so let me know if you would like to have any details on steps of this argument.) 


As a reality check, the standard example of a homotopy equivalence which is not a homeomorphism is obtained by permuting the pages of a book of $I$-bundles. But the boundary of one will immerse in the other, and hence will not satisfy your hypothesis. 


