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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A Brouwer translation theorem for free homeomorphisms
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by Edward E. Slaminka PDF
Trans. Amer. Math. Soc. 306 (1988), 277-291 Request permission

Abstract:

We prove a generalization of the Brouwer Translation Theorem which applies to a class of homeomorphisms (free homeomorphisms) which admit fixed points, but retain a dynamical property of fixed point free orientation preserving homeomorphsims. That is, if $h:{M^2} \to {M^2}$ is a free homeomorphism where ${M^2}$ is a surface, then whenever $D$ is a disc and $h(D) \cap D = \emptyset$, we have that ${h^n}(D) \cap D = \emptyset$ for all $n \ne 0$. Theorem. Let $h$ be a free homeomorphism of ${S^2}$, the two-sphere, with finite fixed point set $F$. Then each $p \in {S^2} - F$ lies in the image of an embedding ${\phi _p}:({R^2}, 0) \to ({S^2} - F, p)$ such that: (i) $h{\phi _p} = {\phi _p}\tau$, where $\tau (z) = z + 1$ is the canonical translation of the plane, and (ii) the image of each vertical line under ${\phi _p}$ is closed in ${S^2} - F$.
References
  • Richard B. Barrar, Proof of the fixed point theorems of Poincaré and Birkhoff, Canadian J. Math. 19 (1967), 333–343. MR 210106, DOI 10.4153/CJM-1967-024-5
  • L. E. J. Brouwer, Continuous one-one transformations of surfaces in themselves, Nederl. Akad. Wetensch. Proc. 11 (1909), 788-798. —, Continuous one-one transformations of surfaces in themselves. II, Nederl. Akad. Wetensch. Proc. 12 (1909), 286-297. —, Continuous one-one transformations of surfaces in themselves. III, Nederl. Akad. Wetensch. Proc. 13 (1911), 300-310. —, Continuous one-one transformations of surfaces in themselves. IV, Nederl. Akad. Wetensch. Proc. 13 (1911), 200-310.
  • L. E. J. Brouwer, Über eineindeutige, stetige Transformationen von Flächen in sich, Math. Ann. 69 (1910), no. 2, 176–180 (German). MR 1511582, DOI 10.1007/BF01456868
  • L. E. J. Brouwer, Zur Analysis Situs, Math. Ann. 68 (1910), no. 3, 422–434 (German). MR 1511570, DOI 10.1007/BF01475781
  • —, Über Abbildungen von Manningfaltigkeiten, Math. Ann. 71 (1911), 97-115.
  • L. E. J. Brouwer, Beweis des ebenen Translationssatzes, Math. Ann. 72 (1912), no. 1, 37–54 (German). MR 1511684, DOI 10.1007/BF01456888
  • —, Remark on the plane translation theorem, Nederl. Akad. Wetensch. Proc. 21 (1919), 935-936. —, Collected works. 2 (H. Freudenthal, ed.), North-Holland, 1976.
  • M. Brown, Homeomorphisms of two-dimensional manifolds, Houston J. Math. 11 (1985), no. 4, 455–469. MR 837985
  • B. de Kerekjarto, The plane translation theorem of Brouwer and the last geometric theorem of Poincaré, Acta Sci. Math. Szeged 4 (1928/1929), 86-102. —, Über die fixpunktfreinen Abbildungen der Eben, Acta Litt. ac Sci. 6 (1934), 226-234. W. Scherrer, Translationen über einfach zusammenhängende Gebeite, Viertelkschr. Naturf. Ges. Zurich 70 (1925), 77-84. E. E. Slaminka, A Brouwer translation theorem for free homeomorphisms (Ph.D. Dissertation), University Microfilms, Ann Arbor, Mich., 1984. E. Sperner, Über die fixpunktfreien Abbildungen der Ebene, Hamburger Math. Eizelschr. 14 (1933), 1-47. H. Terasaka, Ein Beweis des Brouwerschen ebenen Translationssatzes, Japan. J. Math. 7 (1930), 61-69.
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 306 (1988), 277-291
  • MSC: Primary 54H20
  • DOI: https://doi.org/10.1090/S0002-9947-1988-0927691-X
  • MathSciNet review: 927691