Abstract
We show that locally compact abelian p-groups can be embedded in the first Hawaiian group on a compact path connected subspace of the Euclidean space of dimension four. This result gives a new geometric interpretation for the classification of locally compact abelian groups which are rich in commuting closed subgroups. It is then possible to introduce the idea of an algebraic topology for topologically modular locally compact groups via the geometry of the Hawaiian earring. Among other things, we find applications for locally compact groups which are just noncompact.
Acknowledgements
We thank the referee for constructive comments on the original version of the manuscript.
References
[1] H. Abels, Finite Presentability of S-Arithmetic Groups. Compact Presentability of Solvable Groups, Lecture Notes in Math. 1261, Springer, Berlin, 1987. 10.1007/BFb0079708Search in Google Scholar
[2] D. L. Armacost, The Structure of Locally Compact Abelian Groups, Monogr. Textb. Pure Appl. Math. 68, Marcel Dekker, New York, 1981. Search in Google Scholar
[3] A. Babaee, B. Mashayekhy and H. Mirebrahimi, On Hawaiian groups of some topological spaces, Topology Appl. 159 (2012), no. 8, 2043–2051. 10.1016/j.topol.2012.01.013Search in Google Scholar
[4] A. Babaee, B. Mashayekhy, H. Mirebrahimi and H. Torabi, On a van Kampen theorem for Hawaiian groups, Topology Appl. 241 (2018), 252–262. 10.1016/j.topol.2018.04.005Search in Google Scholar
[5] R. Baer, Abelian groups without elements of finite order, Duke Math. J. 3 (1937), no. 1, 68–122. 10.1215/S0012-7094-37-00308-9Search in Google Scholar
[6] O. Bogopolski and A. Zastrow, The word problem for some uncountable groups given by countable words, Topology Appl. 159 (2012), no. 3, 569–586. 10.1016/j.topol.2011.10.003Search in Google Scholar
[7] J. Braconnier, Sur les groupes topologiques localement compacts, Ph.D. thesis, University of Nancy, 1945. Search in Google Scholar
[8] P.-E. Caprace and N. Monod, Decomposing locally compact groups into simple pieces, Math. Proc. Cambridge Philos. Soc. 150 (2011), no. 1, 97–128. 10.1017/S0305004110000368Search in Google Scholar
[9] Y. Cornulier and P. de la Harpe, Metric Geometry of Locally Compact Groups, EMS Tracts Math. 25, European Mathematical Society (EMS), Zürich, 2016. 10.4171/166Search in Google Scholar
[10] K. Eda, A locally simply connected space and fundamental groups of one point unions of cones, Proc. Amer. Math. Soc. 116 (1992), no. 1, 239–249. 10.1090/S0002-9939-1992-1132409-0Search in Google Scholar
[11] K. Eda, Free σ-products and non-commutatively slender groups, J. Algebra 148 (1992), no. 1, 243–263. 10.1016/0021-8693(92)90246-ISearch in Google Scholar
[12] K. Eda, The fundamental groups of one-dimensional wild spaces and the Hawaiian earring, Proc. Amer. Math. Soc. 130 (2002), no. 5, 1515–1522. 10.1090/S0002-9939-01-06431-0Search in Google Scholar
[13] K. Eda and K. Kawamura, Homotopy and homology groups of the n-dimensional Hawaiian earring, Fund. Math. 165 (2000), no. 1, 17–28. 10.4064/fm-165-1-17-28Search in Google Scholar
[14] W. Herfort, K. H. Hofmann and F. G. Russo, Periodic Locally Compact Groups, De Gruyter Stud. Math. 71, De Gruyter, Berlin, 2019. 10.1515/9783110599190Search in Google Scholar
[15] W. Herfort, K. H. Hofmann and F. G. Russo, When is the sum of two closed subgroups closed in a locally compact abelian group?, Topology Appl. 270 (2020), Article ID 106958. 10.1016/j.topol.2019.106958Search in Google Scholar
[16] K. H. Hofmann and S. A. Morris, The Structure of Compact Groups, De Gruyter Stud. Math. 25, De Gruyter, Berlin, 2006. 10.1515/9783110199772Search in Google Scholar
[17] B. Kadri, Characterization of non-compact locally compact groups by cocompact subgroups, J. Group Theory 23 (2020), no. 1, 17–24. 10.1515/jgth-2019-0029Search in Google Scholar
[18] U. H. Karimov and D. Repovš, On noncontractible compacta with trivial homology and homotopy groups, Proc. Amer. Math. Soc. 138 (2010), no. 4, 1525–1531. 10.1090/S0002-9939-09-10217-4Search in Google Scholar
[19] U. H. Karimov and D. Repovš, Hawaiian groups of topological spaces, Uspekhi Mat. Nauk 61 (2006), no. 5(371), 185–186. 10.1070/RM2006v061n05ABEH004363Search in Google Scholar
[20] J. E. Keesling and Y. B. Rudyak, On fundamental groups of compact Hausdorff spaces, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2629–2631. 10.1090/S0002-9939-07-08696-0Search in Google Scholar
[21] C. Kosniowski, A First Course in Algebraic Topology, Cambridge University, Cambridge, 1980. 10.1017/CBO9780511569296Search in Google Scholar
[22] J. Mycielski, Independent sets in topological algebras, Fund. Math. 55 (1964), 139–147. 10.4064/fm-55-2-139-147Search in Google Scholar
[23] J. Pawlikowski, The fundamental group of a compact metric space, Proc. Amer. Math. Soc. 126 (1998), no. 10, 3083–3087. 10.1090/S0002-9939-98-04399-8Search in Google Scholar
[24]
A. J. Przeździecki,
Every countable group is the fundamental group of some compact subspace of
[25] J. J. Rotman, An Introduction to Algebraic Topology, Grad. Texts in Math. 119, Springer, New York, 1988. 10.1007/978-1-4612-4576-6Search in Google Scholar
[26] F. G. Russo, Locally compact groups which are just not compact, Adv. Pure Appl. Math. 1 (2010), no. 2, 285–291. 10.1515/apam.2010.020Search in Google Scholar
[27] S. Shelah, Can the fundamental (homotopy) group of a space be the rationals?, Proc. Amer. Math. Soc. 103 (1988), no. 2, 627–632. 10.1090/S0002-9939-1988-0943095-3Search in Google Scholar
[28] B. de Smit, The fundamental group of the Hawaiian earring is not free, Internat. J. Algebra Comput. 2 (1992), no. 1, 33–37. 10.1142/S0218196792000049Search in Google Scholar
[29] E. Specker, Additive Gruppen von Folgen ganzer Zahlen, Port. Math. 9 (1950), 131–140. 10.1007/978-3-0348-9259-9_5Search in Google Scholar
[30] Ž. Virk, Realizations of countable groups as fundamental groups of compacta, Mediterr. J. Math. 10 (2013), no. 3, 1573–1589. 10.1007/s00009-013-0274-0Search in Google Scholar
[31] J. S. Wilson, Profinite Groups, Oxford University, Oxford, 2002. Search in Google Scholar
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