Arhangel'skii sheaf amalgamations in topological groups

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 α_1.5 is equivalent to Arhangel’skı’s formally stronger property α_1. 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 C_p(X) of continuous real-valued functions on X, with the topology of pointwise convergence, has Arhangel’skı’s property α_1
but is not countably tight. This result follows from results of Arhangel’ski ̆ı–Pytkeev, Moore and Todorcevic, and provides a new solution, with stronger properties than the earlier solution, of a problem of Averbukh and Smolyanov (1968) concerning topological vector spaces.