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Let /: B -» Y be a cofibration whose cofiber is a Moore space. We give necessary and sufficient conditions for/ to be induced by a map of the desuspension of the cofiber into B. These conditions are especially simple if B and Y are nilpotent.We obtain some results on the existence of equivariant Moore spaces, and use them to construct examples of noninduced cofibrations between nilpotent spaces. Our machinery also leads to a cell structure proof of the characterization of pre-nilpotent spaces due to Dror and Dwyer [7], and to a simple proof, for finite fundamental group, of the result of Brown and Kahn [4] that homotopy dimension equals simple cohomological dimension in nilpotent spaces.