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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)
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