Stable splittings, spaces of representations and almost commuting elements in Lie groups

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In this paper the space of almost commuting elements in a Lie group is studied through a homotopical point of view. In particular a stable splitting after one suspension is derived for these spaces and their quotients under conjugation. A complete description for the stable factors appearing in this splitting is provided for compact connected Lie groups of rank one.By using symmetric products, the colimits (ⁿ, SU), (ⁿ,U) and (ⁿ, Sp) are explicitly described as finite products of Eilenberg-MacLane spaces.