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

Saeki, Osamu

doi:10.2140/agt.2006.6.539

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

Osamu Saeki

Algebraic & Geometric Topology 6 (2006) 539–572

DOI: 10.2140/agt.2006.6.539

arXiv: 0903.1733

Abstract

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 $g : (ℝ^{3}, 0) \to (ℝ^{2}, 0)$ with $ℝ^{2}$ being oriented, the algebraic number of cusps appearing in a stable perturbation of $g$ is a local topological invariant of $g$ .

Keywords

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

Mathematical Subject Classification 2000

Primary: 57R45

Secondary: 57R75, 58K60, 58K65

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Publication

Received: 22 September 2005

Accepted: 25 January 2006

Published: 7 April 2006

Authors

Osamu Saeki
Faculty of Mathematics Kyushu University Hakozaki Fukuoka 812-8581 Japan
http://www.math.kyushu-u.ac.jp/~saeki/

Volume 6, issue 2 (2006)