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Maps with only Morin singularities and the Hopf invariant one problem

Published online by Cambridge University Press:  01 November 1998

OSAMU SAEKI  and
KAZUHIRO SAKUMA
OSAMU SAEKI
Affiliation:
Department of Mathematics, Faculty of Science, Hiroshima University, Higashi-Hiroshima 739, Japan; e-mail: [email protected]
KAZUHIRO SAKUMA
Affiliation:
Department of General Education, Kochi National College of Technology, Nankoku-City, Kochi 783, Japan; e-mail: [email protected]
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Abstract

We show that the non-existence of elements in the p-stem πSp of Hopf invariant one implies that: there exists no smooth map f[ratio ]M→N with only fold singularities when M is a closed n-dimensional manifold with odd Euler characteristic and N is an almost parallelizable p-dimensional manifold (n[ges ]p), provided that p≠1, 3, 7. In fact, the result itself is originally due to Kikuchi and Saeki [25, 34]. Our proof clarifies the relationship between the two problems and gives a new insight to the problem of the global singularity theory. Furthermore we generalize the above result to maps with only Morin singularities of types Ak with k[les ]3 when p≠1, 2, 3, 4, 7, 8.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 124 , Issue 3 , November 1998 , pp. 501 - 511
Copyright
© Cambridge Philosophical Society 1998

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