Invariants of Fold-maps via Stable Homotopy Groups

Abstract

In the 2-jet space $J^2(n,p)$ of smooth map germs $({\mathbf{R}}^n,0)\to ({\mathbf{R}}^p,0)$ with $n\ge p\ge 2$, we consider the subspace ${\Omega }^{n-p_{+}1,0}(n,p)$ 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 ${\Omega }^{n-p_{+}1,0}(n,p)$. Let $N$ and $P$ be smooth ($C^{\infty }$) manifolds of dimensions $n$ and $p$. A smooth map $f:N\to P$ 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 ${\lim }_{k,l\to \infty }{\pi }_{n+k+l}(T({\nu }_P^k)\wedge T_{(}{\hat{\gamma }}_{G_{n-p_{+}1,l}}^l))$. Here, ${\nu }_P^k$ is the normal bundle of $P$ in ${\mathbf{R}}^{p+k}$ and ${\hat{\gamma }}_{G_{n-p_{+}1,l}}^l$ denote the canonical vector bundles of dimension $l$ over the grassman manifold $G_{n-p+1,l}$. We also prove the oriented version.