Mathematical Proceedings of the Cambridge Philosophical Society



Maps with only Morin singularities and the Hopf invariant one problem


OSAMU SAEKI a1 1 and KAZUHIRO SAKUMA a2 2
a1 Department of Mathematics, Faculty of Science, Hiroshima University, Higashi-Hiroshima 739, Japan; e-mail: saeki@top2.math.sci.hiroshima-u.ac.jp
a2 Department of General Education, Kochi National College of Technology, Nankoku-City, Kochi 783, Japan; e-mail: sakuma@ge.kochi-ct.ac.jp

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[rightward arrow]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[gt-or-equal, slanted]p), provided that p[not equal]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[less-than-or-eq, slant]3 when p[not equal]1, 2, 3, 4, 7, 8.

(Received April 12 1996)
(Revised November 1 1996)



Footnotes

1 Partially supported by Grant-in-Aid for Encouragement of Young Scientists (No. 08740057), Ministry of Education, Science and Culture, Japan.

2 Partially supported by Grant-in-Aid for Encouragement of Young Scientists (No. 08874004), Ministry of Education, Science and Culture, Japan.