I am trying to prove the following claim which I saw in some paper:
Let $M$ be an $n$-dimensional smooth, oriented, simply connected manifold, which is homeomorphic to a bounded subset of $\mathbb{R^n}$.
Then, there exists an orientation-preserving smooth immersion $f:M\longrightarrow \mathbb{R}^n$.
Remarks:
1) The dimensions of the domain and co-domain are the same, therefore any such $f$ will be in particular a local diffeomorphism.
2) Since $M$ is conected every local diffeomorphism is orientation preserving or orientaton-reversing. Hence, it is enough to find an immersion and then compose it with any orientation reversing diffeomorphism of $\mathbb{R}^n$ if necessary.
3) I think some of the above conditions on $M$ are unnecessary for this to hold (they seem unnatural to me, and are related to other aspects of the paper).