Note on the mappings of complex projective spaces (Q2702787)

Let \(\mathbb CP^s\) denote complex projective \(s\)-space. Let \(n\) be even and \(2k\leq n\). This note contributes to the study of singularities of smooth maps \(\mathbb CP^n \rightarrow \mathbb CP^{n+k-1}\). NEWLINENEWLINENEWLINEThe homotopy classification of (continuous) maps \(\mathbb CP^n \rightarrow \mathbb CP^{n+k-1}\) is given by the integral cohomology group \(H^2(\mathbb CP^n; \mathbb Z)\). Given any smooth map \(f\: \mathbb CP^n \rightarrow \mathbb CP^{n+k-1}\) such that \(f^\ast (\operatorname {generator}) \neq \pm \operatorname {generator}\), it is known [\textit{S. Feder}, Topology 4, 143-158 (1965; Zbl 0151.32301)] that \(f\) must have singular points (hence is not an immersion). It is proved here that not all of its singularities are the simplest possible, and that the same can be said about singularities of maps bordant to a nonzero (rational) multiple of \(f\). NEWLINENEWLINENEWLINEThe proof uses an earlier result on immersions due to the author [Math. Proc. Camb. Philos. Soc. 103, No. 1, 89-95 (1988; Zbl 0644.57015)].