Transverse fundamental group and projected embeddings

الأصل · EN

For a generic degree d smooth map f: Nⁿ -> Mⁿ we introduce its "transverse fundamental group" π(f), which reduces to π₁(M) in the case where f is a covering, and in general admits a monodromy homomorphism π(f) -> S|d|; nevertheless, we show that π(f) can be non-trivial already for rather simple degree 1 maps Sⁿ -> Sⁿ. We apply π(f) to the problem of lifting f to an embedding N -> M x R²: for such a lift to exist, the monodromy π(f) -> S|d| must factor through the group of concordance classes of |d|-component string links. At least if |d|<7, this requires π(f) to be torsion-free.