Characteristic classes and existence of singular maps (Q2839954)

The results of this paper concern topological consequences for the source manifold of the existence of a corank one map of negative codimension. The authors obtain many interesting results in this direction of the following type (Theorem 1.1): Let \(M\) be an \(n\)-dimensional orientable closed manifold admitting a Morin map \(f : M^n \rightarrow S^{n-k}\) into the \((n-k)\)-sphere, where \(k\) is odd. Then for any pair of multiindices \(R=(r_1,\cdots, r_m)\) and \(S=(s_1,\cdots, s_m)\) of any equal length \(m\) and equal degree \(\sum_{j=1}^m r_j=\sum_{j=1}^m s_j\), we have \(w_R(M)=w_S(M)\) if all the entries \(r_j\) and \(s_j\) are at least \(k+3\). When \(M\) admits even a fold map, then the previous result holds even for entries at least \(k+2\).