Folding maps and the surgery theory on manifolds

Abstract

Let $f:N\rightarrow P$ 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 $f$ : $N\rightarrow P$ canonically determines the homotopy class of a bundle map of $TN\oplus\theta_{N}$ to $TP\oplus\theta_{P}$, where $\theta_{N}$ and $\theta_{P}$ are the trivial line bundles over $N$ and $P$ respectively. When $P$ is a closed manifold in addition, we define the set $\Omega_{\mathrm{fold}}(P)$ 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 $\displaystyle \lim_{k\rightarrow\infty}SG_{k}$, 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 $\Omega_{\mathrm{fold}}(P)$ to the set of homotopy classes $[P,SG]$. We relate $\Omega_{\mathrm{fold}}(P)$ with the set of smooth structures on $P$ by applying the surgery theory.