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We give a characterization of closed, simply connected, rationally elliptic6-manifolds in terms of their rational cohomology rings and a partialclassification of their real cohomology rings. We classify rational, real andcomplex homotopy types of closed, simply connected, rationally elliptic7-manifolds. We give partial results in dimensions 8 and 9.

Classification and Characterization of rationally elliptic manifolds in low dimensions