Maps with only Morin singularities and the Hopf invariant one problem

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.

The Chess conjecture

“…In [10] (see also [7]) Saeki and Sakuma describe a remarkable relation between the problem of the existence of certain Morin mappings and the Hopf invariant one problem. Using this relation the authors show that if the Euler characteristic of P is odd, Q is almost parallelizable, and there exists a cusp mapping f : P → Q, then the dimension of Q is 1, 2, 3, 4, 7 or 8.…”

Morin singularities of coframes and frames

“…Therefore, Morin singularities are fundamental and frequently arise as singularities of maps from one manifold to another, as observed by K. Saji in [12]. Morin singularities have been studied by many authors in different contexts as [7,1,4,10,11], and more recently [6,15,18,5,2,12,14,13,9]. In particular, papers of J.M.…”

Characteristic classes and existence of singular maps

“…− There exist fold maps and cusp maps of M into an almost parallelizable manifold only if the Euler characteristic χ(M ) is even, under the assumption that n−k is big enough [SS98]. Refinements of [SS98] include results for Morin maps as well when k is odd [An07,Sad03] but nothing is known when χ(M ) is even. − For odd k , the self-intersection class of the singular set of a generic corank 1 map…”