Theory of Fibered 3-Knots in $S^5$ and its Applications

Journal of Mathematical Sciences

Saeki, Osamu¹ 

关键词: Fibered knot;    Seifert matrix;    isotopy of 4-manifolds;    complex isolated hypersurface singularity;   

DOI  :  

学科分类：数学（综合）

来源: University of Tokyo * Department of Mathematical Sciences

【 摘 要 】

Let$K$beaclosedconnectedorientable3-manifoldembeddedin$S^5$whosecomplementsmoothlyfibersoverthecirclewithsimplyconnectedfibers.Suchanembedded3-manifoldiscalleda{itsimplefibered/}3-{itknot}.Inthispaperwestudysuchembedded3-manifoldsandgivevariousnewresults,whichareclassifiedintothreetypes:(1)thosewhicharesimilartohigherdimensionalfiberedknots,(2)thosewhicharepeculiartofiberedknotsin$S^5$,and(3)applications.Amongtheresultsoftype(1)aretheisotopycriterionsviaSeifertmatrices,determiningfibered3-knotsbytheirexteriors,topologicalorstableuniquenessofthefiberingstructures,andtheeffectivenessofplumbingoperations.Asresultsoftype(2),wegivevariousexplicitexamplesoffibered3-knotswiththesamediffeomorphismtypeoftheabstract3-manifoldsandwithcongruentSeifertmatricesbutwithdifferentisotopytypes.Wealsogivesomeexamplesoffiberedknotswhoseexteriorsarediffeomorphicbutwithdifferentisotopytypes.WealsoshowthatthereexistinfinitelymanyembeddingsofthepuncturedK3surfaceinto$S^5$whicharefibersoftopologicalfibrationsbutwhichcanneverbeafiberofanysmoothfibrations.Weconstructafibered3-knotwhichisdecomposableasaknotsuchthatneitherofthefactorknotsarefibered.Asaresultoftype(3),westudytopologicalisotopiesofhomeomorphismsofsimplyconnected4-manifoldswithboundarybyusingthetechniquesoffibered3-knots.Wealsoapplyourtechniquestotheembeddingproblemofsimplyconnected4-manifoldsinto$S^6$.Finallywegivesomeapplicationstothetopologicalstudyofisolatedhypersurfacesingularitiesin${fC}^3$.

【 授权许可】

Unknown