Arhangel'skiı sheaf amalgamations in topological groups

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We consider amalgamation properties of convergent sequences in topological groups and topological vector spaces. The main result of this paper is that, for arbitrary topological groups, Nyikos's property α₁.₅ is equivalent to Arhangel'skiı's formally stronger property α₁. This result solves a problem of Shakhmatov (2002), and its proof uses a new perturbation argument. We also prove that there is a topological space X such that the space Cₚ(X) of continuous real-valued functions on X, with the topology of pointwise convergence, has Arhangel'skiı's property α₁ but is not countably tight. This result follows from results of Arhangel'skiı--Pytkeev, Moore and Todorčević, and provides a new solution, with remarkable properties, to a problem of Averbukh and Smolyanov (1968) concerning topological vector spaces. The Averbukh--Smolyanov problem was first solved by Plichko (2009), using Banach spaces with weaker locally convex topologies.