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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The group of homeomorphisms of a solenoid


Author: James Keesling
Journal: Trans. Amer. Math. Soc. 172 (1972), 119-131
MSC: Primary 57E05; Secondary 57A15
MathSciNet review: 0315735
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0315735-6
PII: S 0002-9947(1972)0315735-6
Keywords: $ n$-mean, compact abelian topological group, automorphism group, solenoid, group of homeomorphisms
Article copyright: © Copyright 1972 American Mathematical Society