Let $K$ be a closed connected orientable 3-manifold embedded in $S^5$ whose complement smoothly fibers over the circle with simply connected fibers. Such an embedded 3-manifold is called a {\it simple fibered\/} 3-{\it knot}. In this paper we study such embedded 3-manifolds and give various new results, which are classified into three types: (1) those which are similar to higher dimensional fibered knots, (2) those which are peculiar to fibered knots in $S^5$, and (3) applications. Among the results of type (1) are the isotopy criterions via Seifert matrices, determining fibered 3-knots by their exteriors, topological or stable uniqueness of the fibering structures, and the effectiveness of plumbing operations. As results of type (2), we give various explicit examples of fibered 3-knots with the same diffeomorphism type of the abstract 3-manifolds and with congruent Seifert matrices but with different isotopy types. We also give some examples of fibered knots whose exteriors are diffeomorphic but with different isotopy types. We also show that there exist infinitely many embeddings of the punctured K3 surface into $S^5$ which are fibers of topological fibrations but which can never be a fiber of any smooth fibrations. We construct a fibered 3-knot which is decomposable as a knot such that neither of the factor knots are fibered. As a result of type (3), we study topological isotopies of homeomorphisms of simply connected 4-manifolds with boundary by using the techniques of fibered 3-knots. We also apply our techniques to the embedding problem of simply connected 4-manifolds into $S^6$. Finally we give some applications to the topological study of isolated hypersurface singularities in ${\bf C}^3$.
雑誌名
Journal of mathematical sciences, the University of Tokyo
巻
6
号
4
ページ
691 - 756
発行年
1999
ISSN
13405705
書誌レコードID
AA11021653
フォーマット
application/pdf
日本十進分類法
415
Mathematical Reviews Number
MR1742599
Mathmatical Subject Classification
57Q45(MSC1991)
57N35(MSC1991)
出版者
Graduate School of Mathematical Sciences, The University of Tokyo
原稿受領日
1999-03-15
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Theory of Fibered 3-Knots in $S^5$ and its Applications
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