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The goal of this paper is to ameliorate the sufficients conditions, alreadyestablished by the first author so that the sum of the numbers of Betti, of1-connected rational finite CW-complex, is higher than the dimension of his$\mathbb Q$-vectorial space of homotopy, we will present it in two aspects, onealgebraic and another geometrical.

The conjecture H: A lower bound of cohomologic dimension for an elliptic space