Invariants of fold-maps via stable homotopy groups

In the 2-jet space J(2) (n, p) of smooth map germs (R-n, 0) --> (R-p, 0) with n greater than or equal to p greater than or equal to 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-infinity) 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-->infinity) pi(n+k+l)(T(nu(P)(k))boolean ANDT(gamma(Gn-p+1,l)(l))). Here, v(P)(k) is the normal bundle of P in Rp+k and gamma(Gn-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.