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

DOI:
https://doi.org/10.1090/S0002-9939-1988-0955025-9

MathSciNet review:
955025

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Abstract: In this paper it is investigated as to when a nonempty closed subset of a metric product containing intervals or spheres as factors can be the fixed point set of an autohomeomorphism of . It is shown that if is the Hilbert cube or contains either the real line or a -sphere as a factor, then can be any nonempty closed subset. In the case where is in , the interior of the closed unit , a strong necessary condition is given. In particular, for can neither be a nonamphicheiral knot nor a standard closed or nonplanar bordered surface.

**[1]**T. A. Chapman,*Lectures on Hilbert cube manifolds*, American Mathematical Society, Providence, R. I., 1976. Expository lectures from the CBMS Regional Conference held at Guilford College, October 11-15, 1975; Regional Conference Series in Mathematics, No. 28. MR**0423357****[2]**Richard H. Crowell and Ralph H. Fox,*Introduction to knot theory*, Springer-Verlag, New York-Heidelberg, 1977. Reprint of the 1963 original; Graduate Texts in Mathematics, No. 57. MR**0445489****[3]**John R. Martin and William Weiss,*Fixed point sets of metric and nonmetric spaces*, Trans. Amer. Math. Soc.**284**(1984), no. 1, 337–353. MR**742428**, https://doi.org/10.1090/S0002-9947-1984-0742428-1**[4]**William S. Massey,*Algebraic topology: an introduction*, Springer-Verlag, New York-Heidelberg, 1977. Reprint of the 1967 edition; Graduate Texts in Mathematics, Vol. 56. MR**0448331****[5]**Dale Rolfsen,*Knots and links*, Publish or Perish, Inc., Berkeley, Calif., 1976. Mathematics Lecture Series, No. 7. MR**0515288****[6]**Helga Schirmer,*On fixed point sets of homeomorphisms of the 𝑛-ball*, Israel J. Math.**7**(1969), 46–50. MR**0246286**, https://doi.org/10.1007/BF02771745**[7]**Helga Schirmer,*Fixed point sets of homeomorphisms of compact surfaces*, Israel J. Math.**10**(1971), 373–378. MR**0298647**, https://doi.org/10.1007/BF02771655**[8]**-,*Fixed point sets of continuous selfmaps*, Fixed Point Theory Proc. (Sherbrooke, 1980), Lecture Notes in Math., vol. 866, Springer-Verlag, Berlin, 1981, pp. 417-428.**[9]**L. E. Ward Jr.,*Fixed point sets*, Pacific J. Math.**47**(1973), 553–565. MR**0367963**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0955025-9

Keywords:
Fixed point set,
orientation preserving (reversing) autohomeomorphism,
Hilbert cube,
knot,
closed surface,
bordered surface

Article copyright:
© Copyright 1988
American Mathematical Society