## Journal of the Mathematical Society of Japan

- J. Math. Soc. Japan
- Volume 53, Number 2 (2001), 357-382.

### 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 ${\mathrm{lim}}_{k\to \infty}S{G}_{k}$, 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.

#### Article information

**Source**

J. Math. Soc. Japan, Volume 53, Number 2 (2001), 357-382.

**Dates**

First available in Project Euclid: 9 June 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.jmsj/1213023462

**Digital Object Identifier**

doi:10.2969/jmsj/05320357

**Mathematical Reviews number (MathSciNet)**

MR1815139

**Zentralblatt MATH identifier**

0980.58026

**Subjects**

Primary: 58K15: Topological properties of mappings

Secondary: 57R45: Singularities of differentiable mappings 57R67: Surgery obstructions, Wall groups [See also 19J25] 57R55: Differentiable structures 55Q10: Stable homotopy groups

**Keywords**

Folding singularity jet space manifold surgery theory homotopy class

#### Citation

ANDO, Yoshifumi. Folding maps and the surgery theory on manifolds. J. Math. Soc. Japan 53 (2001), no. 2, 357--382. doi:10.2969/jmsj/05320357. https://projecteuclid.org/euclid.jmsj/1213023462