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Fixed point sets of homeomorphisms of metric products

Author: John R. Martin
Journal: Proc. Amer. Math. Soc. 103 (1988), 1293-1298
MSC: Primary 55M20; Secondary 54H25, 57M25, 57N99
MathSciNet review: 955025
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Abstract: In this paper it is investigated as to when a nonempty closed subset $ A$ of a metric product $ X$ containing intervals or spheres as factors can be the fixed point set of an autohomeomorphism of $ X$. It is shown that if $ X$ is the Hilbert cube $ Q$ or contains either the real line $ R$ or a $ (2n - 1)$-sphere $ {S^{2n - 1}}$ as a factor, then $ A$ can be any nonempty closed subset. In the case where $ A$ is in $ \operatorname{Int}({B^{2n + 1}}{\text{)}}$, the interior of the closed unit $ (2n + 1) - {\text{ball }}{B^{2n + 1}},$, a strong necessary condition is given. In particular, for $ {B^3},A$ can neither be a nonamphicheiral knot nor a standard closed or nonplanar bordered surface.

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Keywords: Fixed point set, orientation preserving (reversing) autohomeomorphism, Hilbert cube, knot, closed surface, bordered surface
Article copyright: © Copyright 1988 American Mathematical Society

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