Homotopical Nilpotency and Homology Nilpotency: A Dimension--Connectivity Bound

Let $X$ be a $(q-1)$-connected rational space of finite type, where $q\ge 2$, with homotopical nilpotency $ni{l}_h(X)=n\ge 1$. If $H^{>N}(X;Q)=0$ and $N\le q(n+3)-3$, we prove that $Hnil(X)=n$. More precisely, the sufficient bound is $N\le a_n(M_X)+2q-3$, where $a_n(M_X)$ is the first nonzero internal cohomology degree of $(M_X^{+}{)}^{n+1}$ in the minimal model. A family of examples proves that the constant $-3$ in this refined bound is optimal; optimality of the uniform bound is not asserted. The proof uses two stages of adjoining primitives to an ideal of the fixed minimal model.