## 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

### MSC:

 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