Transverse fundamental group and projected embeddings

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Abstract

For a generic degree d smooth map f: NnMn 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 SnSn. We apply π(f) to the problem of lifting f to an embedding NM × 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.

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© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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