Let p: X → X/A be a quotient map, where A is a subspace of X. We study the conditions under which p ∗(π qtop1 (X, x 0)) is dense in π qtop1 (X/A,∗)), where the fundamental groups have the natural quotient topology inherited from the loop space and p * is a continuous homomorphism induced by the quotient map p. In addition, we present some applications in order to determine the properties of π qtop1 (X/A,∗). In particular, we establish conditions under which π qtop1 (X/A,∗) is an indiscrete topological group.
Similar content being viewed by others
References
A. Arhangelskii and M. Tkachenko, “Topological groups and related structures,” Atlant. Stud. Math. (2008).
D. Biss, “The topological fundamental group and generalized covering spaces,” Topology Its Appl., 124, 355–371 (2002).
J. Brazas, “The topological fundamental group and free topological groups,” Topology Its Appl., 158, 779–802 (2011).
J. Brazas, “ The fundamental group as topological group,” Topology Its Appl., 160, 170–188 (2013).
J. S. Calcut and J. D. McCarthy, “ Discreteness and homogeneity of the topological fundamental group,” Topology Proc., 34, 339–349 (2009).
J. S. Calcut, R. E. Gompf, and J. D. McCarthy, “On fundamental groups of quotient spaces,” Topology Its Appl., 159, 322–330 (2012).
J. W. Cannon and G. R. Conner, “The combinatorial structure of the Hawaiian earring group,” Topology Its Appl., 106, 225–271 (2000).
H. Fischer, D. Repovs, Z. Virk, and A. Zastrow, “On semilocally simply connected spaces,” Topology Its Appl., 158, 397–408 (2011).
H. B. Griffiths, “The fundamental group of two space with a common point,” Quart. J. Math., 5, 175–190 (1954).
B. Mashayekhy, A. Pakdaman, and H. Torabi, “Spanier spaces and covering theory of nonhomotopically path Hausdorff spaces,” Georg. Math. J., 20, 303–317 (1201).
J. Morgan and I. Morrison, “A van Kampen theorem for weak joins,” Proc. London Math. Soc., 53, 562–576 (1986).
J. R. Munkres, Topology: A First Course, 2nd edn., Prentice-Hall, Upper Saddle River, NJ (2000).
A. Pakdaman, H. Torabi, and B. Mashayekhy, “Small loop spaces and covering theory of nonhomotopically Hausdorff spaces,” Topology Its Appl., 158, 803–809 (2011).
A. Pakdaman, H. Torabi, and B. Mashayekhy, “On H-groups and their applications to topological fundamental group,” arXiv:1009.5176v1.
S. Smale, “A Vietoris mapping theorem for homotopy,” Proc. Amer. Math. Soc., 8, 604–610 (1957).
E. H. Spanier, Algebraic Topology, McGraw-Hill, New York (1966).
H. Torabi, A. Pakdaman, and B. Mashayekhy, “Topological fundamental groups and small generated coverings,” Math. Slovaca (to appear).
L. Vietoris, “Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen,” Math. Ann., 97, 454–472 (1927).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 12, pp. 1700–1711, December, 2013.
Rights and permissions
About this article
Cite this article
Torabi, H., Pakdaman, A. & Mashayekhy, B. On the Topological Fundamental Groups of Quotient Spaces. Ukr Math J 65, 1883–1897 (2014). https://doi.org/10.1007/s11253-014-0904-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-014-0904-0