Realization Boundaries

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1 \begin{thebibliography}{9}
2 \addcontentsline{toc}{chapter}{References}
3 \markboth{References}{}
4 \bibitem{Montalban}
5 Antonio Montalb\'an.
6 \emph{Computable Structure Theory: Within the Arithmetic}.
7 Perspectives in Logic, Cambridge University Press, 2021.
8 For the cited numbering, see the author's draft of June 25, 2019, Example III.39 and the following paragraph, p.~50.
9 \url{https://math.berkeley.edu/~antonio/CSTpart1.pdf}.
10 
11 \bibitem{Calvert}
12 Wesley Calvert, Valentina Harizanov, and Alexandra Shlapentokh.
13 Turing degrees of isomorphism types of algebraic objects.
14 \emph{Journal of the London Mathematical Society} 75(2) (2007), 273--286.
15 \href{https://doi.org/10.1112/jlms/jdl012}{doi:10.1112/jlms/jdl012}.
16 Preprint: \href{https://arxiv.org/abs/math/0507128}{arXiv:math/0507128}.
17 
18 \bibitem{Fuchs}
19 L\'aszl\'o Fuchs.
20 \emph{Abelian Groups}.
21 Springer Monographs in Mathematics, Springer, 2015.
22 \href{https://doi.org/10.1007/978-3-319-19422-6}{doi:10.1007/978-3-319-19422-6}.
23 General background for the terminology of pure subgroups and mixed abelian groups. The arguments used here are proved in the text.
24 
25 \bibitem{Lupini}
26 Martino Lupini, Alexander Melnikov, and Andr\'e Nies.
27 \emph{Computable topological abelian groups}.
28 \href{https://arxiv.org/abs/2105.12897v2}{arXiv:2105.12897v2}, 2021.
29 Definition 3.3 supplies the computable inverse-system convention. Section 3.2 describes symmetric cocycles and abelian extensions.
30 
31 \bibitem{Wilkes}
32 Gareth Wilkes.
33 \emph{Part III Profinite Groups}.
34 Lecture notes, Lent Term 2020.
35 Sections 1.2 and 3.1 discuss profinite completion and its functoriality.
36 \url{https://www.dpmms.cam.ac.uk/~grw46/LectureNotes.pdf}.
37 \end{thebibliography}

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