On the Euler-Poincaré characteristics of a simply connected rationally elliptic CW-complex

For a simply connected rationally elliptic CW-complex $X$, we show that the cohomology and the homotopy Euler-Poincaré characteristics are related to two new numerical invariants namely ${\eta }_X$ and ${\rho }_X$ which we define using the Whitehead exact sequences of the Quillen and the Sullivan models of $X$.