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.
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Notes
In [21], Milnor defined the link group \({\mathcal {G}}(L)\) for links L in \(S^3\), which is used for the classification of Brunnian links up to link homotopy. We follow from Birman’s definition of link group as the fundamental group of the link complement [1]. Then we call \({\mathcal {G}}(L)\) as Milnor’s homotopy link group.
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Acknowledgments
The authors would like to thank Joan Birman and Haynes Miller for their encouragements and helpful suggestions on this project. Fuquan Fang and Fengchun Lei supported in part by a Key Grant (No. 11431009) and an Overseas-Collaboration Grant (No. 11329101) of NSFC of China. Research is supported by the Singapore Ministry of Education research Grant (AcRF Tier 1 WBS No. R-146-000-190-112) and a Grant (No. 11329101) of NSFC of China.
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Fang, F., Lei, F. & Wu, J. The Symmetric Commutator Homology of Link Towers and Homotopy Groups of 3-Manifolds. Commun. Math. Stat. 3, 497–526 (2015). https://doi.org/10.1007/s40304-015-0071-0
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DOI: https://doi.org/10.1007/s40304-015-0071-0
Keywords
- Homotopy groups
- Link groups
- Symmetric commutator subgroups
- Intersection subgroups
- Link invariants
- Brunnian-type links
- Strongly non-splittable links