Obstructions to the existence of fold maps

Abstract

We study smooth maps between smooth manifolds with only fold points as their singularities, and clarify the obstructions to the existence of such a map in a given homotopy class for certain dimensions. The obstructions are described in terms of characteristic classes, which arise as Postnikov invariants, and can be interpreted as primary and secondary obstructions to the elimination of certain singularities. We also discuss the relationship between the existence problem of fold maps and that of vector fields of stabilized tangent bundles.

Special generic maps and Gromoll filtration

“…For example, it is known that an orientable closed n-dimensional manifold admits a special generic map into R n if and only if M is stably parallelizable. When the manifold M is nonorientable of dimension n, it admits a special generic map into R n if and only if there exists a set of n nowhere dependent sections of T M ⊕ ε, where T M is the tangent bundle of M and ε is the trivial line bundle over M (see [1] and [13,Corollary 2.4]).…”

Fold maps associated to geodesic random walks on non-positively curved manifolds

“…Conditions for the existence of a fold map between two manifolds have been studied for example in [Sae92], [And04], and [SSS10].…”

Characteristic classes and existence of singular maps

“…− There exists a fold map f : M → Q with cokernel f * T Q/f * df (T M ) being trivial on the singular set if and only if there is a bundle epimorphism T M ⊕ ε 1 → T Q [An04,Sae92]. This gives a complete answer to the problem of existence of fold maps with k ≡ 0 mod 2 [An04], which can be easily used for further computations when k is even, see for example [SSS10]. − More general versions of this result are deep theorems stating h-principles, which are hard to apply directly and led to criteria using Thom polynomials, see for example [An85,An87,An01].…”