On the Spanier Groups and Covering and Semicovering Spaces

2012/07/18 by Hamid Torabi, Torabi, Hamid, Ali Pakdaman +3 · 1 citation
Mathematics · #55Q05 #57M05 #57M10 #57M12 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #math.AT #math.GT #msc:55Q05 #msc:57M05 #msc:57M10 #msc:57M12

paper · pdf · doi:10.48550/arxiv.1207.4394

11 pages,1 figure

arxiv created 2012/07/18 · arxiv updated 2012/07/19

Abstract

For a connected, locally path connected space X, let H be a subgroup of the fundamental group of X, π₁(X,x). We show that there exists an open cover \cal U of X such that H contains the Spanier group π(\U,x) if and only if the core of H in π₁(X,x) is open in the quasitopological fundamental group π₁^qtop(X,x) or equivalently it is open in the topological fundamental group π₁^τ(X,x). As a consequence, using the relation between the Spanier groups and covering spaces, we give a classification for connected covering spaces of X based on the conjugacy classes of subgroups with open core in π₁^qtop(X,x). Finally, we give a necessary and sufficient condition for the existence of a semicovering. Moreover, we present a condition under which every semicovering of X is a covering.

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