Cobordism of exact links
In this paper we introduce the notion of exact links, which constitute a broad class of high dimensional links.
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arXiv subjects
Osamu Saeki
Publications and source records attributed to Osamu Saeki.
Obstructions to the existence of fold maps
We study smooth maps between smooth manifolds with only fold points as their singularities, and clarify the obstructions to the existence of such a map in a given homotopy class for certain dimensions. The obstructions are described in terms of characteristic classes, which arise as Postnikov invariants, and can be interpreted as primary and secondary obstructions to the elimination of certain singularities. We also discuss the relationship between the existence problem of fold maps and that of vector fields of stabilized tangent bundles.
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Cobordism of algebraic knots defined by Brieskorn polynomials
In this paper we study the cobordism of algebraic knots associated with weighted homogeneous polynomials, and in particular Brieskorn polynomials. Under some assumptions we prove that the associated algebraic knots are cobordant if and only if the Brieskorn polynomials have the same exponents.
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Cobordism of Morse functions on surfaces, the universal complex of singular fibers and their application to map germs
We give a new and simple proof for the computation of the oriented and the unoriented fold cobordism groups of Morse functions on surfaces. We also compute similar cobordism groups of Morse functions based on simple stable maps of 3-manifolds into the plane. Furthermore, we show that certain cohomology classes associated with the universal complexes of singular fibers give complete invariants for all these cobordism groups. We also discuss invariants derived from hypercohomologies of the universal homology complexes of singular fibers. Finally, as an application of the theory of universal complexes of singular fibers, we show that for generic smooth map germs g: (R^3, 0) --> (R^2, 0) with R^2 being oriented, the algebraic number of cusps appearing in a stable perturbation of g is a local topological invariant of g.
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Singular fibers of stable maps and signatures of 4-manifolds
We show that for a C^infty stable map of an oriented 4-manifold into a 3-manifold, the algebraic number of singular fibers of a specific type coincides with the signature of the source 4-manifold.
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Regular homotopy classes of immersions of 3-manifolds into 5-space
We give geometric formulae which enable us to detect (completely in some cases) the regular homotopy class of an immersion with trivial normal bundle of a closed oriented 3-manifold into 5-space. These are analogues of the geometric formulae for the Smale invariants due to Ekholm and the second author. As a corollary, we show that two embeddings into 5-space of a closed oriented 3-manifold with no 2-torsion in the second cohomology are regularly homotopic if and only if they have Seifert surfaces with the same signature. We also show that there exist two embeddings $F_0$ and $F_8 : T^3 \hookrightarrow {\bf R}^5$ of the 3-torus $T^3$ with the following properties: (1) $F_0 \sharp h$ is regularly homotopic to $F_8$ for some immersion $h : S^3 \looparrowright {\bf R}^5$, and (2) the immersion $h$ as above cannot be chosen from a regular homotopy class containing an embedding.
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