Invariants of Fold-maps via Stable Homotopy Groups

In the 2-jet space J2(n,p) of smooth map germs (Rn, 0) → (Rp, 0) with n ≥ p ≥ 2, we consider the subspace Ωn-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 Ωn-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 limk,l → ∞πn+k+l(T(vPk) ∧ T (γGn-p+1,ll)). Here, vPk is the normal bundle of P in Rp+k and γGn-p+1,ll denote the canonical vector bundles of dimension l over the grassman manifold Gn-p+l,l. We also prove the oriented version.