Paths in spaces of homeomorphisms on the plane
type: Item
instance of: scholarly article
title: Paths in spaces of homeomorphisms on the plane
zbMATH Open document ID: 1258.54003
published in: Topology and its Applications
publication date: 2012-10-11T00:00:00Z
review text: The path components and connected components are determined for the space \(\mathcal H(\mathbb C)\) of homeomorphisms on the complex plane \(\mathbb C\) when endowed with the pointwise, compact-open and fine topology, respectively (denoted as \( \mathcal H_p(\mathbb C)\), \(\mathcal H_k(\mathbb C)\), and \( \mathcal H_f(\mathbb C)\)); the space \(\mathcal H_u(\mathbb C)\) -- i.e. \( \mathcal H(\mathbb C)\) with the uniform topology -- is also considered, but characterizations of these components are left as open problems. One of the main results shows that \(\mathcal H_k(\mathbb C)\) has exactly two path components, the set of orientation preserving homomorphisms \(\mathcal H^+(\mathbb C)\) and the set of orientation reversing homomorphisms \(\mathcal H^-(\mathbb C)\), moreover, \( \mathcal H^+(\mathbb C)\) is clopen in \( \mathcal H_k(\mathbb C)\) so it is also the connected component of \( \mathcal H_k(\mathbb C)\) containing the identity homeomorphism, and \( \mathcal H^-(\mathbb C)\) is the other connected component of \( \mathcal H_k(\mathbb C)\). For a contrast, \(\mathcal H^+(\mathbb C)\) is dense (with a dense complement \(\mathcal H^-(\mathbb C)\)) in \( \mathcal H_p(\mathbb C)\); in fact, \( \mathcal H_p(\mathbb C)\) is itself pathwise connected. On the other hand, \(\mathcal H_u(\mathbb C)\) and \(\mathcal H_f(\mathbb C)\) have infinitely many path components and connected components. It is also shown that the arc components in each of the spaces \( \mathcal H_p(\mathbb C)\), \(\mathcal H_k(\mathbb C)\), \( \mathcal H_f(\mathbb C)\)), \(\mathcal H_u(\mathbb C)\) are the same as the path components, and at least in each of the spaces \( \mathcal H_p(\mathbb C)\), \(\mathcal H_k(\mathbb C)\), and \( \mathcal H_f(\mathbb C)\)), the connected components are the same as the path components.
reviewed by: Laszlo Zsilinszky
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54D05
zbMATH DE Number: 6092997
zbMATH Keywords: space of homeomorphisms on the plane
zbMATH Keywords: isotopy on the plane
zbMATH Keywords: path
zbMATH Keywords: path component
zbMATH Keywords: arc
zbMATH Keywords: orientation reserving
zbMATH Keywords: orientation reversing
author: R. A. McCoy
MaRDI profile type: Publication
full work available at URL: j.topol.2012.08.016
OpenAlex ID: W2045613247
cites work: Topologies for Homeomorphism Groups
cites work: An Elementary Proof of the Jordan-Schoenflies Theorem
cites work: Q3994935
cites work: A Proof that the Group of all Homeomorphisms of the Plane onto Itself is Locally Arcwise Connected
cites work: The Space of All Self-Homeomorphisms of a Two-Cell which Fix the Cell's Boundary is an Absolute Retract
cites work: Q3562034
cites work: Completeness properties of function spaces
cites work: Q3293867
cites work: Q3138776
cites work: The Jordan-Schonflies Theorem and the Classification of Surface
cites work: Q3812929
DOI: 10.1016/J.TOPOL.2012.08.016
description: scientific article; zbMATH DE number 6092997