Commuting homeomorphisms of \(S^ 2\)
type: Item
instance of: scholarly article
title: Commuting homeomorphisms of \(S^ 2\)
zbMATH Open document ID: 0755.57012
DOI: 10.1016/0040-9383(92)90022-A
author: Michael Handel
published in: Topology
publication date: 1993-01-16T00:00:00Z
review text: It was proven by \textit{C. Bonatti} [Ann. Math., II. Ser. 129, 61-69 (1989; Zbl 0689.57019)] that, if \(f,g\in \text{Diff}^ 1_ +(S^ 2)\) commute and are sufficiently \(C^ 1\) close to the identity, then they have a common fixed point. This answered a question of H. Rosenberg concerning existence of closed leaves in a certain class of foliations. In this paper, the author, after observing that commuting diffeomorphisms of \(S^ 2\) do not generally have common fixed points, gives a sufficient condition that they do, which generalizes Bonatti's theorem. If \(f,g \in \text{Diff}^ 1_ +(S^ 2)\) commute, he defines a kind of ``winding number'' \(W(f,g) \in \mathbb{Z}_ 2\) and proves that the vanishing of \(W(f,g)\) implies the existence of a common fixed point for \(f\) and \(g\). In fact, \(W(f,g)\) is defined for commuting \(f,g\in \text{Homeo}_ +(S^ 2)\) and, if \(f\) and \(g\) each have finite fixed point sets, the author proves that \(W(f,g) = 0\) implies that \(f\) and \(g\) have a common fixed point. Moreover, if at least one of \(f\) and \(g\) has more than one fixed point, then they have in common at least two fixed points. These are the main theorems and related results are proven for commuting homeomorphisms of \(\mathbb{R}^ 2\) and of closed orientable surfaces.
reviewed by: Lawrence Conlon
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 57N05
zbMATH DE Number: 89835
zbMATH Keywords: winding number
zbMATH Keywords: commuting diffeomorphisms of \(S^ 2\)
zbMATH Keywords: common fixed points
zbMATH Keywords: closed orientable surfaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 89835