On the equivariant Hopf theorem
type: Item
instance of: scholarly article
title: On the equivariant Hopf theorem
zbMATH Open document ID: 1017.55005
DOI: 10.1016/S0040-9383(02)00015-0
author: Davide L. Ferrario
published in: Topology
publication date: 2003-03-11T00:00:00Z
review text: Hopf's classification theorem states that the homotopy set \([X, S^n]\) of maps from a compact \(n\)-dimensional manifold \(X\) to \(S^n\) is in one-to-one correspondence with \(H^n(X;\mathbb{Z})\). The aim of this paper is to give an equivariant version of this theorem for a finite group \(G\). Let \(X\) be a compact locally smooth \(G\)-manifold and \(S\) an orthogonal \(G\)-sphere such that for fixed manifolds \(X^H\) and \(S^H\dim X^H\leq \dim S^H\) holds and \(S^K\nsubseteq S^H\) if \(H\nsubseteq K\) for isotropy subgroups \(H\) and \(K\) for \(X\). An essential point for classifying \(G\)-maps here is to use equivariant obstruction theory. In order to apply this method the author introduces an unstable equivariant degree \(\widetilde{d}_G\). For each isotropy group \(H\) let \(X_H\subset X^H\) denote the subspace consisting of points \(x\in X\) with isotropy \(H\) and let \(C\) be a component of \(X_H\). Then the above two conditions on \(X\) and \(S\) allow to choose a point \(y_H\in S^H\) with minimal isotropy type. Moreover, for any \(G\)-map \(f :X\to G\) one finds \((f^H)^{-1}y_H\) to be compact in \(C\) whenever \(\dim C= \dim S^H\). So assign to \(\widetilde{d}_G(f)(C)\) the Brouwer degree of \(f^H|C\) in this case and zero otherwise. Then, to put it simply, \(\widetilde{d}_G(f)\) is defined as a collection of \(d_G(f)(C)s\). This unstable degree is rather fine as compared with the stable one due to tom Dieck and others. If \(X^H\) is orientable for every isotropy group \(H\), then these two degrees are equivalent in a class of certain groups named 2-split groups (Proposition 5.2). Let \(p_H: X_H\to X_H/W_G H\) be the projection onto the quotient by the Weyl group of \(H\) in \(G\). Let \(\overline{C}=p_H(C)\) where \(C\) is as above. This is called the full-dimensional chamber of \(X/G\) (with isotropy type \((H)\)) whenever \(\dim\overline{C} >0\) and \(\dim \overline{C} = \dim S^H\). To state the main results of this paper one needs further the notio…
reviewed by: Haruo Minami
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 55Q05
Mathematics Subject Classification ID: 55S37
Mathematics Subject Classification ID: 55S91
Mathematics Subject Classification ID: 55P92
zbMATH DE Number: 1880022
zbMATH Keywords: equivariant Hopf theorem
zbMATH Keywords: equivariant degree
zbMATH Keywords: classification of equivariant maps
zbMATH Keywords: Bredon-Illman cohomology group
zbMATH Keywords: equivariant obstruction theory
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1880022