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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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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.
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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