Images of finite maps with one-dimensional double point set
type: Item
instance of: scholarly article
title: Images of finite maps with one-dimensional double point set
zbMATH Open document ID: 0943.57012
DOI: 10.1016/S0166-8641(97)00209-5
author: Kevin Houston
published in: Topology and its Applications
publication date: 1999-06-24T00:00:00Z
review text: The Goryunov spectral sequence [\textit{V. Goryunov} and \textit{D. Mond}, Compos. Math. 89, No. 1, 45-80 (1993; Zbl 0839.32017)] is used to calculate the homology of the image of finite-to-one proper maps. In particular a formula for the Euler characteristic of the image of such a map is given in terms of the Euler characteristics of the multiple point spaces of the map. The \(k\)th multiple point space \({\mathcal D}^k(f)\) of a map \(f:X\to Y\) is the closure of the set \[ \{(x_1,\dots, x_k)\in X^k\mid f(x_1)= \dots= f(x_k),\;x_i\neq x_j\}. \] As a corollary the Izumiya-Marar formula [\textit{S. Izumiya} and \textit{W. L. Marar}, J. Geom. 52, No. 1-2, 108-119 (1995; Zbl 0840.57010)] for the Euler characteristic of the image \(f(N)\) of a closed surface in a 3-manifold is obtained.
Mathematics Subject Classification ID: 57N65
Mathematics Subject Classification ID: 58C99
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 1307890
zbMATH Keywords: alternating homology
zbMATH Keywords: singularities of a map
zbMATH Keywords: Goryunov spectral sequence
zbMATH Keywords: Euler characteristic
zbMATH Keywords: Izumiya-Marar formula
reviewed by: Alexander Dranishnikov
MaRDI profile type: Publication
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full work available at URL: s0166-8641(97)00209-5
OpenAlex ID: W2033547956
description: scientific article; zbMATH DE number 1307890