Cobordism of Morse functions on surfaces, the universal complex of singular fibers and their application to map germs

Saeki, Osamu¹

¹Faculty of Mathematics, Kyushu University, Hakozaki, Fukuoka 812-8581, Japan

Algebraic and Geometric Topology, Tome 6 (2006) no. 2, pp. 539-572

Résumé

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: (ℝ3,0) → (ℝ2,0) with ℝ2 being oriented, the algebraic number of cusps appearing in a stable perturbation of g is a local topological invariant of g.

Détail

DOI : 10.2140/agt.2006.6.539

Keywords: Morse function, cobordism, singular fiber, universal complex, simple stable map, hypercohomology, stable perturbation, map germ

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