TY - JOUR

T1 - Cobordism of Morse maps and its application to map germs

AU - Ikegami, Kazuichi

AU - Saeki, Osamu

N1 - Funding Information:
The second author has been partially supported by Grant-in-Aid for Scientific Research (No. 19340018), Japan Society for the Promotion of Science.
Copyright:
Copyright 2009 Elsevier B.V., All rights reserved.

PY - 2009/7

Y1 - 2009/7

N2 - 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.

AB - 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.

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U2 - 10.1017/S0305004109002321

DO - 10.1017/S0305004109002321

M3 - Article

AN - SCOPUS:68349144652

VL - 147

SP - 235

EP - 254

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 1

ER -