A simple proof of the degree formula for \((\mathbb{Z}/p)\)-equivariant maps
type: Item
instance of: scholarly article
title: A simple proof of the degree formula for \((\mathbb{Z}/p)\)-equivariant maps
zbMATH Open document ID: 0790.55003
DOI: 10.1007/BF02571658
published in: Mathematische Zeitschrift
publication date: 1994-04-12T00:00:00Z
full work available at URL: https://eudml.org/doc/174487
review text: The paper contains elementary and short proofs of several Borsuk-Ulam type theorems. Here are two examples. 1. If \(G= \mathbb{Z}/p\), \(p\) a prime, acts on a normed linear space via linear isometries and if \(f: \overline{\Omega}\subset X\to X\) is a compact perturbation of the identity which is also equivariant then \(\deg(f, \Omega,0)\equiv \deg(f^ G, \Omega^ G,0)\bmod p\). Here \(f^ G\) denotes the restriction of \(f\) to the set of points fixed under \(G\) and it is assumed that these Leray-Schauder degrees are defined. 2. If \(V\) and \(W\) are orthogonal representations of \(G= (\mathbb{Z}/p)^ k\) such that \(V^ G=0= W^ G\) then the existence of an equivariant map \(SV\to SW\) implies \(\dim W\geq \dim V\). Here \(SV\) is the unit sphere of \(V\). There are related results about Lusternik-Schnirelmann category of lens spaces and about the equivariant category of representation spheres. Similar results hold for \(G=(S^ 1)^ k\). Published proofs of these theorems usually apply methods from algebraic topology, in particular if \(G\) is different from \(\mathbb{Z}/2\) or \(S^ 1\). Here only properties of the Leray-Schauder degree and Sard's lemma are being used.
Mathematics Subject Classification ID: 55M25
Mathematics Subject Classification ID: 55M30
zbMATH DE Number: 549733
zbMATH Keywords: Borsuk-Ulam theorems
zbMATH Keywords: degree of equivariant maps
zbMATH Keywords: Lusternik-Schnirelmann category
zbMATH Keywords: equivariant Lusternik-Schnirelmann category
author: Thomas J. Bartsch
MaRDI profile type: Publication
cites work: On the genus of representation spheres
cites work: On the existence of Borsuk-Ulam theorems
cites work: Q5185568
cites work: A Borsuk-Ulam theorem for orthogonal \(T^ k\) and \(Z^ r_ p\) actions and applications
cites work: Mappings of manifolds and the notion of degree
cites work: The classification of 𝐺-spaces
OpenAlex ID: W1997492492
description: scientific article; zbMATH DE number 549733