Brown, M.; Kister, J. M. Invariance of complementary domains of a fixed point set. (English) Zbl 0547.57010 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. Cited in 15 Documents 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 PDF BibTeX XML Cite \textit{M. Brown} and \textit{J. M. Kister}, Proc. Am. Math. Soc. 91, 503--504 (1984; Zbl 0547.57010) Full Text: DOI OpenURL