The group of homeomorphisms of a solenoid
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- by James Keesling PDF
- Trans. Amer. Math. Soc. 172 (1972), 119-131 Request permission
Abstract:
Let $X$ be a topological space. An $n$-mean on $X$ is a continuous function $\mu :{X^n} \to X$ which is symmetric and idempotent. In the first part of this paper it is shown that if $X$ is a compact connected abelian topological group, then $X$ admits an $n$-mean if and only if ${H^1}(X,Z)$ is $n$-divisible where ${H^m}(X,Z)$ is $m$-dimensional Čech cohomology with integers $Z$ as coefficient group. This result is used to show that if ${\Sigma _a}$ is a solenoid and $\operatorname {Aut} ({\Sigma _a})$ is the group of topological group automorphisms of ${\Sigma _a}$, then $\operatorname {Aut} ({\Sigma _a})$ is algebraically ${Z_2} \times G$ where $G$ is $\{ 0\} ,{Z^n}$, or $\oplus _{i = 1}^\infty Z$. For a given ${\Sigma _a}$, the structure of $\operatorname {Aut} ({\Sigma _a})$ is determined by the $n$-means which ${\Sigma _a}$. admits. Topologically, $\operatorname {Aut} ({\Sigma _a})$ is a discrete space which has two points or is countably infinite. The main result of the paper gives the precise topological structure of the group of homeomorphisms $G({\Sigma _a})$ of a solenoid ${\Sigma _a}$ with the compact open topology. In the last section of the paper it is shown that $G({\Sigma _a})$ is homeomorphic to ${\Sigma _a} \times {l_2} \times \operatorname {Aut} ({\Sigma _a})$ where ${l_2}$ is separable infinite-dimensional Hilbert space. The proof of this result uses recent results in infinite-dimensional topology and some techniques using flows developed by the author in a previous paper.References
- Georg Aumann, Über Räume mit Mittelbildungen, Math. Ann. 119 (1944), 210–215 (German). MR 12219, DOI 10.1007/BF01563741
- C. Bessaga and A. Pełczyński, A topological proof that every separable Banach space is homeomorphic to a countable product of lines, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 (1969), 487–493 (English, with Russian summary). MR 257995
- B. Eckmann, Räume mit Mittelbildungen, Comment. Math. Helv. 28 (1954), 329–340 (German). MR 65920, DOI 10.1007/BF02566939
- B. Eckmann, T. Ganea, and P. J. Hilton, Generalized means, Studies in mathematical analysis and related topics, Stanford Univ. Press, Stanford, Calif., 1962, pp. 82–92. MR 0169239
- Andrew M. Gleason and Richard S. Palais, On a class of transformation groups, Amer. J. Math. 79 (1957), 631–648. MR 89367, DOI 10.2307/2372567
- David W. Henderson, Infinite-dimensional manifolds are open subsets of Hilbert space, Topology 9 (1970), 25–33. MR 250342, DOI 10.1016/0040-9383(70)90046-7
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
- James Keesling, Using flows to construct Hilbert space factors of function spaces, Trans. Amer. Math. Soc. 161 (1971), 1–24. MR 283751, DOI 10.1090/S0002-9947-1971-0283751-8
- James Keesling, Topological groups whose underlying spaces are separable Fréchet manifolds, Pacific J. Math. 44 (1973), 181–189. MR 377967
- Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104 L. A. Pontrjagin, Continuous groups, GITTL, Moscow, 1954; English transl., Topological groups, Gordon and Breach, New York, 1966. MR 17, 171; MR 34 #1439.
- Wladimiro Scheffer, Maps between topological groups that are homotopic to homomorphisms, Proc. Amer. Math. Soc. 33 (1972), 562–567. MR 301130, DOI 10.1090/S0002-9939-1972-0301130-8
- Kermit Sigmon, Acyclicity of compact means, Michigan Math. J. 16 (1969), 111–115. MR 259899
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
- Norman E. Steenrod, Universal Homology Groups, Amer. J. Math. 58 (1936), no. 4, 661–701. MR 1507191, DOI 10.2307/2371239
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 172 (1972), 119-131
- MSC: Primary 57E05; Secondary 57A15
- DOI: https://doi.org/10.1090/S0002-9947-1972-0315735-6
- MathSciNet review: 0315735