Folding maps and the surgery theory on manifolds

Abstract

Let $f:N\to P$ 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 $f$ : $N\to P$ 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 ${\Omega }_{\mathrm{fold}}\left(P\right)$ 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 ${\lim }_{k\to \infty }S{G}_k$, where $S{G}_k$ denotes the space of all homotopy equivalences of $S^{k-1}$ of degree 1. We prove that there exists an important map of ${\Omega }_{\mathrm{fold}}\left(P\right)$ to the set of homotopy classes $[P,SG]$. We relate ${\Omega }_{\mathrm{fold}}\left(P\right)$ with the set of smooth structures on $P$ by applying the surgery theory.