Regular homotopy classes of immersed surfaces
type: Item
instance of: scholarly article
title: Regular homotopy classes of immersed surfaces
zbMATH Open document ID: 0583.57020
DOI: 10.1016/0040-9383(85)90013-8
author: Ulrich Pinkall
published in: Topology
publication date: 1985-01-01T00:00:00Z
review text: There are two relevant equivalence relations on the set of immersions from a manifold \(M^ m\) into Euclidean space \({\mathbb{R}}^ n:\) regular homotopy and a second coming from the action of diffeomorphisms of M on it. Imposing both relations defines the term ''immersed manifold''. This paper achieves a nice geometrical classification of immersions of a closed surface into \({\mathbb{R}}^ 3\), as well as of immersed surfaces. rmmersions are distinguished by a \({\mathbb{Z}}/4\)-valued (linking number) quadratic form on \(H_ 1(M; {\mathbb{Z}}/2)\). The action of the diffeomorphism group of M is reflected by that of the automorphism group of the form. Quadratic forms are classified by dimension and a \({\mathbb{Z}}/8\)-valued Arf invariant. The outcome is a description of the monoid (by connected sum) of immersed surfaces by generators (torus, ''twisted torus'', Boy surface and its ''mirror image'') and relations. Furthermore it is shown that bordism of immersed surfaces is completely classified by the Arf invariant.
Mathematics Subject Classification ID: 57R42
Mathematics Subject Classification ID: 11E16
zbMATH DE Number: 3934049
zbMATH Keywords: regular homotopy
zbMATH Keywords: immersions of a closed surface into \({\mathbb{R}}^ 3\)
zbMATH Keywords: linking number
zbMATH Keywords: Quadratic forms
zbMATH Keywords: Arf invariant
zbMATH Keywords: bordism of immersed surfaces
reviewed by: Martin Raussen
MaRDI profile type: Publication
full work available at URL: 0040-9383(85)90013-8
OpenAlex ID: W2042059393
description: scientific article; zbMATH DE number 3934049