The group of homeomorphisms of a solenoid
Author:
James Keesling
Journal:
Trans. Amer. Math. Soc. 172 (1972), 119131
MSC:
Primary 57E05; Secondary 57A15
MathSciNet review:
0315735
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Abstract: Let be a topological space. An mean on is a continuous function which is symmetric and idempotent. In the first part of this paper it is shown that if is a compact connected abelian topological group, then admits an mean if and only if is divisible where is dimensional Čech cohomology with integers as coefficient group. This result is used to show that if is a solenoid and is the group of topological group automorphisms of , then is algebraically where is , or . For a given , the structure of is determined by the means which . admits. Topologically, 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 of a solenoid with the compact open topology. In the last section of the paper it is shown that is homeomorphic to where is separable infinitedimensional Hilbert space. The proof of this result uses recent results in infinitedimensional 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/S00029947197203157356
PII:
S 00029947(1972)03157356
Keywords:
mean,
compact abelian topological group,
automorphism group,
solenoid,
group of homeomorphisms
Article copyright:
© Copyright 1972
American Mathematical Society
