The Symmetric Commutator Homology of Link Towers and Homotopy Groups of 3-Manifolds

Abstract

A link tower is a sequence of links with the structure given by removing the last components. Given a link tower, we prove that there is a chain complex consisting of (non-abelian) groups given by the symmetric commutator subgroup of the normal closures in the link group of the meridians excluding the meridian of the last component with the differential induced by removing the last component. Moreover, the homology groups of these naturally constructed chain complexes are isomorphic to the homotopy groups of the manifold M under certain hypothesis. These chain complexes have canonical quotient abelian chain complexes in Minor’s homotopy link groups with their homologies detecting certain differences of the homotopy link groups in the towers.

Keywords

Homotopy groups / Link groups / Symmetric commutator subgroups / Intersection subgroups / Link invariants / Brunnian-type links / Strongly non-splittable links

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Funding

National Natural Science Foundation of China(11329101)

National Natural Science Foundation of China(11329101)

Ministry of Education - Singapore(R-146-000-190-112)

National Natural Science Foundation of China(11431009)

National Natural Science Foundation of China(11431009)