## Invariance of complementary domains of a fixed point set.(English)Zbl 0547.57010

The following useful result seems not to be in the literature. It has a simple but perhaps nonobvious proof. Proposition. Let f be a homeomorphism of a connected topological manifold M with fixed point set F. Then either (1) f is invariant on each (connected) component of $$M-F$$ or (2) there are exactly two components and f interchanges them.

### MSC:

 57N15 Topology of the Euclidean $$n$$-space, $$n$$-manifolds ($$4 \leq n \leq \infty$$) (MSC2010) 57S17 Finite transformation groups 54H20 Topological dynamics (MSC2010)
Full Text: