On the triple points of singular maps
type: Item
instance of: scholarly article
title: On the triple points of singular maps
zbMATH Open document ID: 1009.57035
author: Tobias Ekholm
author: A. Szücs
published in: Commentarii Mathematici Helvetici
publication date: 2002-09-25T00:00:00Z
review text: 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 \(\bar w_{2k_1 + 1}\cdots \bar w_{2k_r + 1}[M]=0\).
reviewed by: Július Korbaš
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 58K30
Mathematics Subject Classification ID: 57R45
Mathematics Subject Classification ID: 57R42
zbMATH DE Number: 1807868
zbMATH Keywords: triple point
zbMATH Keywords: generic (stable) map
zbMATH Keywords: singular map
zbMATH Keywords: immersion
zbMATH Keywords: Stiefel-Whitney number
zbMATH Keywords: normal Stiefel-Whitney class
zbMATH Keywords: linking number
zbMATH Keywords: Pontryagin number
zbMATH Keywords: closed manifold
MaRDI profile type: Publication
full work available at URL: s00014-002-8347-x
OpenAlex ID: W2067892429
DOI: 10.1007/S00014-002-8347-X
description: scientific article; zbMATH DE number 1807868