Folding maps and the surgery theory on manifolds

Abstract

Let f:N→P be a smooth map between n-dimensional oriented manifolds which has only folding singularities. Such a map is called a folding map. We prove that a folding map f : N→P canonically determines the homotopy class of a bundle map of TN⊕θN to TP⊕θP, where θN and θP are the trivial line bundles over N and P respectively. When P is a closed manifold in addition, we define the set Ωfold(P) of all cobordism classes of folding maps of closed manifolds into P of degree 1 under a certain cobordism equivalence. Let SG denote the space limk→∞SGk, where SGk denotes the space of all homotopy equivalences of Sk-1 of degree 1. We prove that there exists an important map of Ωfold(P) to the set of homotopy classes [P,SG]. We relate Ωfold(P) with the set of smooth structures on P by applying the surgery theory.