On Topologized Fundamental Groups and Covering Groups of Topological Groups
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RESEARCH PAPER
On Topologized Fundamental Groups and Covering Groups of Topological Groups Hamid Torabi1 Received: 7 March 2020 / Accepted: 5 October 2020 Ó Shiraz University 2020
Abstract We show that every topological group is a strong small loop transfer space at the identity element. This implies that for a ee equipped with the quotient topology connected locally path connected topological group G, the universal path space G induced by the compact-open topology on P(G, e) is a topological group. Moreover, we prove that there is a one-to-one correspondence between the equivalence classes of connected covering groups of G and the subgroups of p1 ðG; eÞ that contain i ðp1 ðU; eÞÞ for some open neighborhood U of the identity element e. Keywords Topological group Quasitopological fundamental group Covering map
1 Introduction The fundamental group endowed with the quotient topology induced by the natural surjective map q : XðX; x0 Þ ! p1 ðX; x0 Þ, where XðX; x0 Þ is the loop space of ðX; x0 Þ with the compact-open topology, becomes a quasitopological group (see Calcut and McCarthy 2009), and it is denoted by pqtop 1 ðX; x0 Þ. Calcut and McCarthy (2009) showed that the quasitopological fundamental group of a connected locally path connected space X is discrete if and only if X is semilocally 1-connected. On the other hand, Torabi et al. (2015) proved that every nonempty open subset of pqtop 1 ðX; x0 Þ is a disjoint union of some cosets of the small generated subgroup of p1 ðX; x0 Þ. In this paper, we show that if fUj jj 2 Jg is an open basis for a connected locally path connected topological group G at e, then fgi p1 ðUj ; eÞjj 2 Jg is an open basis for pqtop 1 ðG; eÞ at g for every g 2 p1 ðG; eÞ, where i : Uj ! G is the inclusion map. Spanier (1966, Theorem 13 on page 82) introduced a different topology on the fundamental group, which is called the whisker topology by Brodskiy et al. (2012b). This fundamental group is denoted by pwh 1 ðX; x0 Þ, which is & Hamid Torabi [email protected] 1
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
not even a quasitopological group, in general. In this paper, we show that pwh 1 ðG; eG Þ is a topological group when G is a topological group with the identity element eG . Through the paper, for every group G, we denote the identity element of G by eG . Recall that for any pointed topological space ðX; x0 Þ, the whisker topology on the set p1 ðX; x0 Þ is defined by the collection of all the following sets as a basis: ½ai p1 ðU; x0 Þ ¼f½b 2 p1 ðX; x0 Þ j b ’ a d for some loop d : I ! Ug; where ½a 2 p1 ðX; x0 Þ and U is an open neighborhood of x0 (see Abdullahi Rashid et al. 2017 Lemma 3.1). The concept of small loop transfer space introduced and studied by Brodskiy et al. (2012b) is defined below. Definition 1 A topological space X is called a small loop transfer (SLT for short) space at x0 , if for every path a in X with að0Þ ¼ x0 and for every neighborhood U of x0 , there is a neighbo
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