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...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 themeridians excluding themeridian 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.展开更多
基金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)an Overseas-Collaboration Grant(No.11329101)of NSFC of ChinaResearch 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.
文摘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 themeridians excluding themeridian 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.
