Remarks on the topology of folds

Abstract

Abstract. We give some necessary conditions for a closed manifold to admit a smooth map into an Euclidean space with only fold singular points.

Fold maps on 4-manifolds

“…Thus, our result is a generalization of these two, and gives a complete answer to the existence problem of fold maps on closed orientable 4-manifolds. (Note that Sakuma [36] had conjectured that closed orientable 4-manifolds of odd Euler characteristics cannot admit a fold map into R 3 (see also [16,34]). Our result shows that the conjecture is false.)…”

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.…”

Maps with only Morin singularities and the Hopf invariant one problem

“…What kind of geometry is hidden behind this phenomenon? Recall that Kikuchi and Saeki [25] have shown the theorem above by using a result of Brown [8] which states that if a manifold which is not null cobordant immerses into R p in codimension one, then it must be cobordant to RP 0 , RP 2 or RP 6 . Our first purpose of the present paper is to give a new proof of Theorem 1•1, making its geometric features clearer.…”