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

Brown, M.; Kister, J. M.

Proc. Am. Math. Soc. 91, 503-504 (1984).

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)

Keywords:

complementary domains of the fixed point set; homeomorphism of a connected topological manifold