# 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*(

*νPk*) ∧

*T*(

*γ*^

*l__G__n−p*+1,

*l*)). Here,

*νPk*is the normal bundle of

*P*in ℝ_p_+

*k*and

*γ*^

*l__G__n−p*+1,

*l*) denote the canonical vector bundles of dimension

*l*over the grassman manifold

*G__n−p*+1,

*l*. We also prove the oriented version.