Algebraic & Geometric Topology

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

Osamu Saeki

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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:(3,0)(2,0) with 2 being oriented, the algebraic number of cusps appearing in a stable perturbation of g is a local topological invariant of g.

Article information

Source
Algebr. Geom. Topol., Volume 6, Number 2 (2006), 539-572.

Dates
Received: 22 September 2005
Accepted: 25 January 2006
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796534

Digital Object Identifier
doi:10.2140/agt.2006.6.539

Mathematical Reviews number (MathSciNet)
MR2220688

Zentralblatt MATH identifier
1098.57017

Subjects
Primary: 57R45: Singularities of differentiable mappings
Secondary: 57R75: O- and SO-cobordism 58K60: Deformation of singularities 58K65: Topological invariants

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

Citation

Saeki, Osamu. Cobordism of Morse functions on surfaces, the universal complex of singular fibers and their application to map germs. Algebr. Geom. Topol. 6 (2006), no. 2, 539--572. doi:10.2140/agt.2006.6.539. https://projecteuclid.org/euclid.agt/1513796534


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