Homeomorphisms of two-dimensional manifolds. (English) Zbl 0605.57005

The author introduces and develops some theory for the class of free homeomorphisms: A homeomorphism f of a connected manifold \(M^ m\) is called free provided that whenever D is an m-disk in M and \(f(D)\cap D=\emptyset\) then \(f^ n(D)\cap D=\emptyset\) for all positive integers n. In the 2-dimensional case a lemma of L. E. J. Brouwer is used to show (Corollary 5.8) that this class of homeomorphisms includes the fixed point free orientation-preserving homeomorphisms of \({\mathbb{R}}^ 2.\)
More generally, on an orientable 2-manifold it is shown that a homeomorphism is free if and only if it is orientation-preserving and satisfies Brouwer’s translation arc property (Corollary 4.5 and Theorem 4.9) - this means that whenever \(\alpha\) is an arc from p to \(q=f(p)\) in M and \(f(\alpha -\{q\})\cap (\alpha -\{q\})=\emptyset\) then \(f^ n(\alpha -\{q\})\cap (\alpha -\{q\})=\emptyset\) for all \(n>1\). Some further properties and examples of free homeomorphisms on 2-manifolds are also given. Finally, it is shown that the only free homeomorphism on a manifold of dimension bigger than 2 is the identity homeomorphism (Theorem 6.1).
Reviewer: A.Miller


57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57S30 Discontinuous groups of transformations
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
57R50 Differential topological aspects of diffeomorphisms