On the space of equivariant local maps
type: Item
instance of: scholarly article
title: On the space of equivariant local maps
zbMATH Open document ID: 1368.55008
DOI: 10.12775/TMNA.2015.012
author: Piotr Bartłomiejczyk
published in: Topological Methods in Nonlinear Analysis
publication date: 2017-04-04T00:00:00Z
review text: The author's abstract reads: ``We introduce the space of equivariant local maps and present the full proof of the splitting theorem for the set of otopy classes of local maps in the case where we have a representation of a compact Lie group.'' The author's bibliography contains several papers which mention the word otopy. The paper first defining otopy is \textit{D. H. Gottlieb} and \textit{G. Samaranayake} [New York J. Math. 1, 130--148 (1995; Zbl 0883.57025)]. Otopy is a modification of homotopy given by eliminating the ``hom'' from homotopy, and the leading O representing open domains. (Some topologists thought the word was too ugly to use.) The author hopes ``it is worth pointing out that the main motivation for studying the space of local maps is that it forms the natural environment for topological degree theories.'' This is certainly true of local vector fields, where the local index defines the otopy vector field classes. Suppose \(V\) is a vector field on a compact manifold \(M\) with or without boundary \(\partial M\). There is a formula, originally due to \textit{M. Morse} [Am. J. Math. 51, 165--178 (1929; JFM 55.0972.02)], which was not as widely known as it should be: So let \(V\) be a continuous vector field on \(M\) with no zeros on \(\partial M\). Then \(\mathrm{Ind}(V)+\mathrm{Ind}(\partial-V)=\xi(M)\) (my notation) where \(\partial-V\) is the vector field \(V\) restricted to \(\partial M\) and projected onto \(\partial-M\) which is the portion of \(\partial M\) where \(V\) points inside. The local vector field \(\partial-V\) changes by an otopy as \(V\) changes by a homotopy.
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 57R25
zbMATH DE Number: 6700685
zbMATH Keywords: otopy
zbMATH Keywords: local maps
reviewed by: Daniel H. Gottlieb
MaRDI profile type: Publication
full work available at URL: tmna.2015.012
OpenAlex ID: W2472932465
Recommended article: The index of discontinuous vector fields
Recommended article: Proper gradient otopies
Recommended article: The exponential law for partial, local and proper maps and its application to otopy theory
Recommended article: Construction of smooth sphere maps with given degree and a generalization of Morse index formula for smooth vector fields
Recommended article: Fixed point indices and manifolds with collars
Recommended article: Index classes for vector fields and their relation to certain characteristic classes
Recommended article: Gradient otopies of gradient local maps
Recommended article: Q3790319
Recommended article: Q3172162
Recommended article: The Hopf theorem for gradient local vector fields on manifolds
description: scientific article; zbMATH DE number 6700685