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Transactions of the American Mathematical Society

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

 
 

 

Computable topological groups and Pontryagin duality


Author: Alexander Melnikov
Journal: Trans. Amer. Math. Soc. 370 (2018), 8709-8737
MSC (2010): Primary 03D45, 03D80, 43A40
DOI: https://doi.org/10.1090/tran/7355
Published electronically: May 3, 2018
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Abstract: The well-known Pontryagin Duality (classically) reduces the study of compact abelian groups to the algebraic theory of discrete abelian groups. At first glance, Pontryagin Duality seems to be ``algorithmic'' in nature. Quite unexpectedly, the situation is more intricate. Nonetheless, using methods of computable analysis from the work of Weihrauch and modern techniques of computable algebra (e.g., the recent metatheorem), we establish a partial algorithmic analogy of Pontryagin Duality and use it to derive a handful of corollaries. We believe that most of these consequences are fundamental to the emerging systematic theory of computable Polish groups. We also apply our techniques to measure the complexity of the classification problem for profinite and connected compact Polish groups.


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Additional Information

Alexander Melnikov
Affiliation: Institute of Natural and Mathematical Sciences, Massey University, Auckland, New Zealand
Email: alexander.g.melnikov@gmail.com

DOI: https://doi.org/10.1090/tran/7355
Received by editor(s): May 4, 2017
Received by editor(s) in revised form: June 29, 2017, and July 3, 2017
Published electronically: May 3, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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