Transverse fundamental group and projected embeddings

For a generic degree $d$ smooth map $f:N^n\to M^n$ we introduce its “transverse fundamental group” $\pi (f)$, which reduces to $\pi _1(M)$ in the case where $f$ is a covering, and in general admits a monodromy homomorphism $\pi (f)\to S_{|d|}$; nevertheless, we show that $\pi (f)$ can be nontrivial even for rather simple degree $1$ maps $S^n\to S^n$. We apply $\pi (f)$ to the problem of lifting $f$ to an embedding $N\hookrightarrow M\times \mathbb R^2$: for such a lift to exist, the monodromy $\pi (f)\to S_{|d|}$ must factor through the group of concordance classes of $|d|$-component string links. At least if $|d|<7$, this requires $\pi (f)$ to be torsion-free.