Singularities of projected immersions revisited (Q834385)

A generic map \(f:M \to N\) is called a projected immersion if there exists a generic immersion \(g : M \looparrowright N \times \mathbb{R}\), with \( f = \pi \circ g\), where \( \pi : N \times \mathbb{R} \to N\) is the projection. \textit{A. Szűcs} [Bull. Lond. Math. Soc. 32, No. 3, 364--374 (2000; Zbl 1021.57013)] considered projected immersions with \(N\) a Euclidean space and proved that the \(r\)-tuple-point manifold of \(g\) is cobordant to the \(\sum^{{1}_{r-1}}\)-point manifold of \(f\), by showing that the characteristic numbers of the two manifolds involved are equal. In this work, \(N\) is any smooth manifold. The author introduces for \(1 \leq i \leq r\) an auxiliary manifold \(\Lambda^{i}_{r}\) and a map \( \lambda^{i}_{r} : \Lambda^{i}_{r} \to M\) and by an effective geometric construction of a bordism between \( \lambda^{i}_{r}\) and \( \lambda^{i+1}_{r}\) proves that for \(f:M^n \longrightarrow N^{n+k}\) and \(r \geq 1\) one has \(\lambda^{1}_{r} \sim \lambda^{2}_{r} \sim \dots \sim \lambda^{r}_{r}\) in \(\mathfrak{N}_*(M)\). Since \( \lambda^{1}_{r}\) and \( \lambda^{r}_{r}\) are the natural inclusion \( \sum^{{1}_{r-1}}(f) \hookrightarrow M\) and \( g_r : \triangle_r(g) \to M\), respectively, then it follows that \(g_r\) and \(\sum^{{1}_{r-1}}(f) \hookrightarrow M \) are bordant in \(\mathfrak{N}_*(M)\).