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.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 12, pp. 1700–1711, December, 2013.
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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
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DOI: https://doi.org/10.1007/s11253-014-0904-0