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Cobordism of Morse maps and its application to map germs

Published online by Cambridge University Press:  01 July 2009

KAZUICHI IKEGAMI  and
OSAMU SAEKI
KAZUICHI IKEGAMI
Affiliation:
Faculty of Engineering, Hiroshima Institute of Technology, 2-1-1 Miyake, Saeki-ku, Hiroshima 731-5193, Japan. e-mail: [email protected]
OSAMU SAEKI
Affiliation:
Faculty of Mathematics, Kyushu University, Hakozaki, Fukuoka 812-8581, Japan. e-mail: [email protected]
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Abstract

Let f: M → S1 be a Morse map of a closed manifold M into the circle, where a Morse map is a smooth map with only nondegenerate critical points. In this paper, we classify such maps up to fold cobordism. In the course of the classification, we get several fold cobordism invariants for such Morse maps. We also consider a slightly general situation where the source manifold M has boundary and the map f restricted to the boundary has no critical points. Let g: (Rm, 0) → (R2, 0), m ≥ 2, be a generic smooth map germ, where the target R2 is oriented. Using the above-mentioned fold cobordism invariants, we show that the number of cusps with a prescribed index appearing in a C∞ stable perturbation of g, counted with signs, gives a topological invariant of g.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 1 , July 2009 , pp. 235 - 254
Copyright
Copyright © Cambridge Philosophical Society 2009

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