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Auslander-Reiten triangles and quivers are introduced into algebraic topology. It is proved that the existence of Auslander-Reiten triangles characterizes Poincaré duality spaces, and that the Auslander-Reiten quiver is a weak homotopy invariant. The theory is applied to spheres whose Auslander-Reiten triangles and quivers are computed. The Auslander-Reiten quiver over the