Let be a smooth map between -dimensional oriented manifolds which has only folding singularities. Such a map is called a folding map. We prove that a folding map : canonically determines the homotopy class of a bundle map of to , where and are the trivial line bundles over and respectively. When is a closed manifold in addition, we define the set of all cobordism classes of folding maps of closed manifolds into of degree 1 under a certain cobordism equivalence. Let denote the space , where denotes the space of all homotopy equivalences of of degree 1. We prove that there exists an important map of to the set of homotopy classes . We relate with the set of smooth structures on by applying the surgery theory.
Yoshifumi ANDO. "Folding maps and the surgery theory on manifolds." J. Math. Soc. Japan 53 (2) 357 - 382, April, 2001. https://doi.org/10.2969/jmsj/05320357