Invariants of fold-maps via stable homotopy groups (Q1848506)

Let \(N\) and \(P\) be smooth manifolds, \(\dim N \geq \dim P \geq 2\) and \(\dim N- \dim P\) even. The paper deals with smooth maps \(f:N \rightarrow P\) with at most fold singularities, i.e. such that for every point \(q \in N\) there are suitable local coordinate systems \((N,x)\) and \((P,y)\) having \(q\) and \(f(q)\) as origins such that \[ (x_1, \dots x_n) \mapsto (x_1, \dots x_p) \text{ or } (x_1, \dots x_n) \mapsto (x_1, \dots x_{p-1}, \pm x_p^2 \pm \dots \pm x^2_n). \] The goal is to construct a surjection of the set of cobordism classes of such mappings to \(P\) (\(N\) is allowed to vary) onto certain stable homotopy groups. The crucial point is to determine the homotopy type of the space of two-jets of maps having only fold singularities.