Transverse fundamental group and projected embeddings

Abstract

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 nontrivial even for rather simple degree 1 maps Sn → Sn. We apply π(f) to the problem of lifting f to an embedding N ↪ M × 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.