Research Article
BibTex RIS Cite
Year 2020, Volume: 49 Issue: 3, 935 - 949, 02.06.2020
https://doi.org/10.15672/hujms.464056

Abstract

References

  • [1] M. Abdullahi Rashid, B. Mashayekhy, H. Torabi and S.Z. Pashaei, On subgroups of topologized fundamental groups and generalized coverings, Bull. Iranian Math. Soc. 43 (7), 2349–2370, 2017.
  • [2] A. Arhangelskii and M. Tkachenko, Topological Groups and Related Structures, At- lantis Studies in Mathematics, 1. Atlantis Press, Paris, 2008.
  • [3] D. Biss, The topological fundamental group and generalized covering spaces, Topology Appl. 124, 355–371, 2002. (Retraction published 2017, Topology Appl. 217, p. 116)
  • [4] W. Bogley, A. Sierdaski, Universal path spaces, http://people.oregonstate.edu/ ~bogleyw/research/ups.pdf, unpublished preprint.
  • [5] J. Brazas, Semicoverings: a generalization of covering space theory, Homology Ho- motopy Appl. 14, 33–63, 2012.
  • [6] J. Brazas, The fundamental group as topological group, Topology Appl. 160, 170–188, 2013.
  • [7] J. Brazas, Semicoverings, coverings, overlays and open subgroups of the quasitopological fundamental group, Topology Proc. 44, 285-313, 2014.
  • [8] J. Brazas, Generalized covering space theories, Theory Appl. Categ. 30, 1132–1162, 2015.
  • [9] J. Brazas and P. Fabel, Thick Spanier groups and the first shape group, Rocky Moun- tain J. Math. 44, 1415–1444, 2014.
  • [10] J. Brazas and P. Fabel, On fundamental group with the quotient topology, J. Homotopy Relat. Struct. 10, 71–91, 2015.
  • [11] N. Brodskiy, J. Dydak, B. Labuz and A. Mitra, Covering maps for locally path con- nected spaces, Fund. Math. 10, 13–46, 2008.
  • [12] N. Brodskiy, J. Dydak, B. Labuz and A. Mitra, Topological and uniform structures on universal covering spaces, arXiv:1206.0071v1.
  • [13] J. Calcut and J. McCarthy, Discreteness and homogeneity of the topological funda- mental group, Topology Proc. 34, 339–349, 2009.
  • [14] J. Dugundji, A topologized fundamental group, Proc. Nat. Acad. Sci. U. S. A. 36, 141–143, 1950.
  • [15] H. Fischer, D. Repovs, Z. Virk and A. Zastrow, On semilocally simply connected spaces, Topology Appl. 155, 1089–1097, 2008.
  • [16] H. Fischer and A. Zastrow, Generalized universal covering spaces and the shape group, Fund. Math. 197, 167–196, 2007.
  • [17] H. Fischer and A. Zastrow, A core-free semicovering of the Hawaiian Earring, Topol- ogy Appl. 160, 1957–1967, 2013.
  • [18] W. Hurewicz, Homotopie, homologie und lokaler zusammenhang, Fund. Math. 25, 467–485, 1935.
  • [19] N. Jamali, B. Mashayekhy, H. Torabi, S.Z. Pashaei and M. Abdullahi Rashid, On topologized fundamental groups with small loop transfer viewpoints, Acta Math. Viet- namica, 44, 711–722, 2019.
  • [20] M. Kowkabi, H. Torabi and B. Mashayekhy, When is a local homeomorphism a semi- covering map?, Acta Math. Vietnamica, 42, 653–663, 2017.
  • [21] B. Mashayekhy, A. Pakdaman and H. Torabi, Spanier spaces and covering theory of non-homotopically path Hausdorff spaces, Georgian Math. J. 20, 303–317, 2013.
  • [22] A. Pakdaman, H. Torabi and B. Mashayekhy, On the existence of categorical universal coverings, Italian Journal of Pure and Applied Mathematics, 37, 289–300, 2017.
  • [23] S.Z. Pashaei, B. Mashayekhy, H. Torabi and M. Abdullahi Rashid, On strong small loop transfer spaces relative to subgroups of fundamental groups, Topology Appl. 254, 117–131, 2019.
  • [24] E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
  • [25] H. Torabi, A. Pakdaman and B. Mashayekhy, Topological fundamental groups and small generated coverings, Math. Slovaca, 65, 1153–1164, 2015.
  • [26] H. Torabi, A. Pakdaman and B. Mashayekhy, On the Spanier groups and covering and semicovering spaces, arXiv:1207.4394v1.
  • [27] Z. Virk and A. Zastrow, The comparison of topologies related to various concepts of generalized covering spaces, Topology Appl. 170, 52–62, 2014.
  • [28] J, Wilkins, The revised and uniform fundamental groups and universal covers of ge- odesic spaces, Topology Appl. 160, 812–835, 2013.

On subgroup topologies on fundamental groups

Year 2020, Volume: 49 Issue: 3, 935 - 949, 02.06.2020
https://doi.org/10.15672/hujms.464056

Abstract

It is important to classify covering subgroups of the fundamental group of a topological space using their topological properties in the topologized fundamental group. In this paper, we introduce and study some topologies on the fundamental group and use them to classify coverings, semicoverings, and generalized coverings of a topological space. To do this, we use the concept of subgroup topology on a group and discuss their properties. In particular, we explore which of these topologies make the fundamental group a topological group. Moreover, we provide some examples of topological spaces to compare topologies of fundamental groups.

References

  • [1] M. Abdullahi Rashid, B. Mashayekhy, H. Torabi and S.Z. Pashaei, On subgroups of topologized fundamental groups and generalized coverings, Bull. Iranian Math. Soc. 43 (7), 2349–2370, 2017.
  • [2] A. Arhangelskii and M. Tkachenko, Topological Groups and Related Structures, At- lantis Studies in Mathematics, 1. Atlantis Press, Paris, 2008.
  • [3] D. Biss, The topological fundamental group and generalized covering spaces, Topology Appl. 124, 355–371, 2002. (Retraction published 2017, Topology Appl. 217, p. 116)
  • [4] W. Bogley, A. Sierdaski, Universal path spaces, http://people.oregonstate.edu/ ~bogleyw/research/ups.pdf, unpublished preprint.
  • [5] J. Brazas, Semicoverings: a generalization of covering space theory, Homology Ho- motopy Appl. 14, 33–63, 2012.
  • [6] J. Brazas, The fundamental group as topological group, Topology Appl. 160, 170–188, 2013.
  • [7] J. Brazas, Semicoverings, coverings, overlays and open subgroups of the quasitopological fundamental group, Topology Proc. 44, 285-313, 2014.
  • [8] J. Brazas, Generalized covering space theories, Theory Appl. Categ. 30, 1132–1162, 2015.
  • [9] J. Brazas and P. Fabel, Thick Spanier groups and the first shape group, Rocky Moun- tain J. Math. 44, 1415–1444, 2014.
  • [10] J. Brazas and P. Fabel, On fundamental group with the quotient topology, J. Homotopy Relat. Struct. 10, 71–91, 2015.
  • [11] N. Brodskiy, J. Dydak, B. Labuz and A. Mitra, Covering maps for locally path con- nected spaces, Fund. Math. 10, 13–46, 2008.
  • [12] N. Brodskiy, J. Dydak, B. Labuz and A. Mitra, Topological and uniform structures on universal covering spaces, arXiv:1206.0071v1.
  • [13] J. Calcut and J. McCarthy, Discreteness and homogeneity of the topological funda- mental group, Topology Proc. 34, 339–349, 2009.
  • [14] J. Dugundji, A topologized fundamental group, Proc. Nat. Acad. Sci. U. S. A. 36, 141–143, 1950.
  • [15] H. Fischer, D. Repovs, Z. Virk and A. Zastrow, On semilocally simply connected spaces, Topology Appl. 155, 1089–1097, 2008.
  • [16] H. Fischer and A. Zastrow, Generalized universal covering spaces and the shape group, Fund. Math. 197, 167–196, 2007.
  • [17] H. Fischer and A. Zastrow, A core-free semicovering of the Hawaiian Earring, Topol- ogy Appl. 160, 1957–1967, 2013.
  • [18] W. Hurewicz, Homotopie, homologie und lokaler zusammenhang, Fund. Math. 25, 467–485, 1935.
  • [19] N. Jamali, B. Mashayekhy, H. Torabi, S.Z. Pashaei and M. Abdullahi Rashid, On topologized fundamental groups with small loop transfer viewpoints, Acta Math. Viet- namica, 44, 711–722, 2019.
  • [20] M. Kowkabi, H. Torabi and B. Mashayekhy, When is a local homeomorphism a semi- covering map?, Acta Math. Vietnamica, 42, 653–663, 2017.
  • [21] B. Mashayekhy, A. Pakdaman and H. Torabi, Spanier spaces and covering theory of non-homotopically path Hausdorff spaces, Georgian Math. J. 20, 303–317, 2013.
  • [22] A. Pakdaman, H. Torabi and B. Mashayekhy, On the existence of categorical universal coverings, Italian Journal of Pure and Applied Mathematics, 37, 289–300, 2017.
  • [23] S.Z. Pashaei, B. Mashayekhy, H. Torabi and M. Abdullahi Rashid, On strong small loop transfer spaces relative to subgroups of fundamental groups, Topology Appl. 254, 117–131, 2019.
  • [24] E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
  • [25] H. Torabi, A. Pakdaman and B. Mashayekhy, Topological fundamental groups and small generated coverings, Math. Slovaca, 65, 1153–1164, 2015.
  • [26] H. Torabi, A. Pakdaman and B. Mashayekhy, On the Spanier groups and covering and semicovering spaces, arXiv:1207.4394v1.
  • [27] Z. Virk and A. Zastrow, The comparison of topologies related to various concepts of generalized covering spaces, Topology Appl. 170, 52–62, 2014.
  • [28] J, Wilkins, The revised and uniform fundamental groups and universal covers of ge- odesic spaces, Topology Appl. 160, 812–835, 2013.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Mehdi Abdullahi Rashid This is me 0000-0001-7222-3922

Noorollah Jamali This is me 0000-0002-3811-1721

Behrooz Mahsayekhy This is me 0000-0001-5243-0641

Seyyed Zeynal Pashaei This is me 0000-0001-7401-6599

Hamid Torabı 0000-0002-2609-4872

Publication Date June 2, 2020
Published in Issue Year 2020 Volume: 49 Issue: 3

Cite

APA Rashid, M. A., Jamali, N., Mahsayekhy, B., Pashaei, S. Z., et al. (2020). On subgroup topologies on fundamental groups. Hacettepe Journal of Mathematics and Statistics, 49(3), 935-949. https://doi.org/10.15672/hujms.464056
AMA Rashid MA, Jamali N, Mahsayekhy B, Pashaei SZ, Torabı H. On subgroup topologies on fundamental groups. Hacettepe Journal of Mathematics and Statistics. June 2020;49(3):935-949. doi:10.15672/hujms.464056
Chicago Rashid, Mehdi Abdullahi, Noorollah Jamali, Behrooz Mahsayekhy, Seyyed Zeynal Pashaei, and Hamid Torabı. “On Subgroup Topologies on Fundamental Groups”. Hacettepe Journal of Mathematics and Statistics 49, no. 3 (June 2020): 935-49. https://doi.org/10.15672/hujms.464056.
EndNote Rashid MA, Jamali N, Mahsayekhy B, Pashaei SZ, Torabı H (June 1, 2020) On subgroup topologies on fundamental groups. Hacettepe Journal of Mathematics and Statistics 49 3 935–949.
IEEE M. A. Rashid, N. Jamali, B. Mahsayekhy, S. Z. Pashaei, and H. Torabı, “On subgroup topologies on fundamental groups”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 3, pp. 935–949, 2020, doi: 10.15672/hujms.464056.
ISNAD Rashid, Mehdi Abdullahi et al. “On Subgroup Topologies on Fundamental Groups”. Hacettepe Journal of Mathematics and Statistics 49/3 (June 2020), 935-949. https://doi.org/10.15672/hujms.464056.
JAMA Rashid MA, Jamali N, Mahsayekhy B, Pashaei SZ, Torabı H. On subgroup topologies on fundamental groups. Hacettepe Journal of Mathematics and Statistics. 2020;49:935–949.
MLA Rashid, Mehdi Abdullahi et al. “On Subgroup Topologies on Fundamental Groups”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 3, 2020, pp. 935-49, doi:10.15672/hujms.464056.
Vancouver Rashid MA, Jamali N, Mahsayekhy B, Pashaei SZ, Torabı H. On subgroup topologies on fundamental groups. Hacettepe Journal of Mathematics and Statistics. 2020;49(3):935-49.