Automorphisms and embeddings of surfaces and quadruple points of regular homotopies
type: Item
instance of: scholarly article
title: Automorphisms and embeddings of surfaces and quadruple points of regular homotopies
zbMATH Open document ID: 1032.57030
author: Tahl Nowik
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: The author studies a regular homotopy invariant \(Q(\cdot,\cdot)\) of immersions. Let \(i,\;i':F\to \mathbb{R}^3\) be two generic immersions of an orientable closed surface \(F\) to 3-dimensional Euclidean space \(\mathbb{R}^3\), having a generic regular homotopy between them. Then the number \(Q(i,i') \mod 2\) of quadruple points of this regular homotopy is well-defined by Theorem 3.9 of the author's paper [Topology 39, 1069-1088 (2000; Zbl 0962.57015)]. This article contains two main theorems (Theorems 5.6 and 7.3) concerning the invariant \(Q(i,i')\). The first one is the following. For any generic immersion \(i:F\to \mathbb{R}^3\) of an orientable closed surface \(F\) to \(\mathbb{R}^3\) and any diffeomorphism \(h\) of \(F\) such that \(i\) and \(i\circ h\) are generically regularly homotopic, it holds that \(Q(i,i\circ h)=\text{rank }(h_*-Id)+(g(F)+1)\epsilon (h) \mod 2\), where \(h_*\) is the isomorphism of \(H_1(F;\mathbb{Z}_2)\) defined by \(h\), \(g(F)\) denotes the genus of \(F\) and \(\epsilon (f)=0\) or \(1\) according \(h\) being orientation-preserving or not. The second one gives an explicit formula for \(Q(e,e')\) for two generically regularly homotopic embeddings \(e,e':F\to \mathbb{R}^3\). The expression of the formula is really simple. But its calculation may not be as easy as that of the first one, because to determine the first term of the right hand side of the formula, readers are required to construct so called totally singular decompositions of \(H_1(F;\mathbb{Z}_2)\), associated with the embeddings \(e\) and \(e'\), and to determine the rank of the endomorphism \(T-Id\) of \(H_1(F;\mathbb{Z})_2\), where \(T\) is some orthogonal map with respect to the quadratic form induced from the embedding \(e\) and with properties connected to the above decompositions (see \S 6).
reviewed by: Tsutomu Yasui
Mathematics Subject Classification ID: 57R42
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 57R40
zbMATH DE Number: 1782694
zbMATH Keywords: immersion
zbMATH Keywords: embedding
zbMATH Keywords: regular-homotopy
zbMATH Keywords: quadruple-point
zbMATH Keywords: quadratic-form
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1090348354
OpenAlex ID: W1599854321
Wikidata QID: Q115202498
DOI: 10.4310/JDG/1090348354
description: scientific article; zbMATH DE number 1782694