On the triple points of singular maps
Given a closed smooth \(2n\)-dimensional manifold \(M\) and a generic (stable) map \(f: M\rightarrow \mathbb R^{3n}\), let \(t_2(f)\) be the mod \(2\) number of the triple points of \(f\), and let \(l_2(f)\) be the mod \(2\) linking number of \(f\). The authors prove that then \(t_2(f)+l_2(f)= \bar w_n^2[M] + \bar w_{n+1}\bar w_{n-1}[M],\) where \(\bar w_i\in H^i(M; \mathbb Z_2)\) denotes the \(i\)-th normal Stiefel-Whitney class of \(M\), and \([M]\) is the fundamental homology class of \(M\). This result corrects an erroneous formula of \textit{A. Szűcs} [Comment. Math. Helv. 61, 360-369 (1986; Zbl 0602.57021), p. 361]. Another result of the paper under review is the following theorem which, as commented in the paper, resulted from a correspondence between R. Stong and the second author: If \(M\) is an oriented \(4k\)-dimensional manifold and \((2k_1 + 1)+\dots + (2k_r + 1)=4k\), then \(\bar w_{2k_1 + 1}\cdots \bar w_{2k_r + 1}[M]=0\).
scientific article; zbMATH DE number 1807868