  For a generic degree d smooth map f: N^n -> 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) -> S_{|d|};
nevertheless, we show that \pi(f) can be non-trivial already for rather simple
degree 1 maps S^n -> S^n.
  We apply \pi(f) to the problem of lifting f to an embedding N -> M x R^2: for
such a lift to exist, the monodromy \pi(f) -> 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.
