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The twisted sum splits if it is equivalent to $0\\to Y\\to Y \\times Z \\to Z\\to 0$. \\par We study the class $\\ST$ of topological groups $G$ for which every twisted sum $0\\to \\T\\to X \\to G\\to 0$ splits. We prove that this class contains locally precompact groups, sequential direct limits of locally compact groups and topological groups with $\\mathcal{L}_\\infty$ topologies. 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Bello, Mar\\'ia Jes\\'us Chasco, Xabier Dom\\'inguez","submitted_at":"2013-05-20T13:04:17Z","abstract_excerpt":"A twisted sum in the category of topological abelian groups is a short exact sequence $0\\to Y\\to X \\to Z\\to 0$ where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to $0\\to Y\\to Y \\times Z \\to Z\\to 0$. \\par We study the class $\\ST$ of topological groups $G$ for which every twisted sum $0\\to \\T\\to X \\to G\\to 0$ splits. We prove that this class contains locally precompact groups, sequential direct limits of locally compact groups and topological groups with $\\mathcal{L}_\\infty$ topologies. 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