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 TNoplusθ_N to TPoplusθ_P, where θ_N and θ_P are the trivial line bundles over N and P respectively. When P is a closed manifold in addition, we define the set Ω_fo1d(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 \displaystyle \lim_k→∞SG_k, where SG_k denotes the space of all homotopy equivalences of S^k-1 of degree 1. We prove that there exists an important map of Ω_fo1d(P) to the set of homotopy classes [P, SG]. We relate Ω_fo1d(P) with the set of smooth structures on P by applying the surgery theory.