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

J. Math. Sci. Univ. Tokyo
Vol. 6 (1999), No. 4, Page 691--756.

Saeki, Osamu
Theory of Fibered 3-Knots in $S^5$ and its Applications
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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 ${\bf C}^3$.

Keywords: Fibered knot, Seifert matrix, isotopy of 4-manifolds, complex isolated hypersurface singularity

Mathematics Subject Classification (1991): Primary 57Q45; Secondary 57N35
Mathematical Reviews Number: MR1742599

Received: 1999-03-15