Whitney proved that any n-dimensional manifold $M^{n}$ can be embedded into $\mathbb{R}^{2n}$ (as a codimension $n$ submanifold). A similar argument can be used to show that any $n$-dimensional CW complex also embeds into $\mathbb{R}^{2n}$.
I am interested in the following related question. Suppose $M^{2n}$ is a $2n$-dimensional manifold with boundary. When can it embed into $\mathbb{R}^{2n}$?
There are two obvious necessary conditions. First, $M^{2n}$ must be parallelizable (since $\mathbb{R}^{2n}$ is). Second, the intersection form on $M$ must vanish. I will further assume that $M^{2n}$ has the homotopy type of an $(n-1)$-dimensional CW complex, so that Whitney's theorem allows this CW complex to embed into $\mathbb{R}^{2n}$. I believe that these conditions are sufficient: any parallelizable $M^{2n}$ with boundary and homotopy type of an $n-1$-dimensional CW complex embeds into $\mathbb{R}^{2n}$.
Are there any existing results in this direction?
Clarification: I mean that the manifolds are smooth and compact with boundary, and embeddings are smooth.