# Invariants of Fold-maps via Stable Homotopy Groups

### Yoshifumi Ando

Yamaguchi University, Yamaguchi City, Japan

## Abstract

In the 2-jet space $J_{2}(n,p)$ of smooth map germs $(R_{n},0)→(R_{p},0)$ with $n≥p≥2$, we consider the subspace $Ω_{n−p_{+}1,0}(n,p)$ consisting of all 2-jets of regular germs and map germs with fold singularities. In this paper we determine the homotopy type of the space $Ω_{n−p_{+}1,0}(n,p)$. Let $N$ and $P$ be smooth ($C_{∞}$) manifolds of dimensions $n$ and $p$. A smooth map $f:N→P$ is called a fold-map if $f$ has only fold singularities. We will prove that this homotopy type is very useful in ﬁnding invariants of fold-maps. For instance, by applying the homotopy principle for fold-maps in [An3] and [An4] we prove that if $n−p+1$ is odd and $P$ is connected, then there exists a surjection of the set of cobordism classes of fold-maps into $P$ to the stable homotopy group $lim_{k,l→∞}π_{n+k+l}(T(ν_{P})∧T_{(}γ _{G_{n−p1,l}}))$. Here, $ν_{P}$ is the normal bundle of $P$ in $R_{p+k}$ and $γ _{G_{n−p1,l}}$ denote the canonical vector bundles of dimension $l$ over the grassman manifold $G_{n−p+1,l}$. We also prove the oriented version.

## Cite this article

Yoshifumi Ando, Invariants of Fold-maps via Stable Homotopy Groups. Publ. Res. Inst. Math. Sci. 38 (2002), no. 2, pp. 397–450

DOI 10.2977/PRIMS/1145476344