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) → (\mathbf 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^k) \wedge T_(\widehat γ^l_{G_{n−p_+1,l}})) . Here, ν_P^k is the normal bundle of P in \mathbf R^{p+k} and \widehat 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.