Regularity in the growth of the loop space homology of a finite CW complex

To any path connected topological space X X , such that rk ⁡ H i ( X ) > ∞ \operatorname {rk} H_i(X) >\infty for all i ≥ 0 i\geq 0 , are associated the following two sequences of integers: b i = rk ⁡ H i ( Ω X ) b_i= \operatorname {rk} H_i(\Omega X) and r i = rk ⁡ π i + 1 ( X ) r_i= \operatorname {rk} \pi _{i+1}(X) . If X X is simply connected, the Milnor-Moore theorem together with the Poincaré-Birkoff-Witt theorem provides an explicit relation between these two sequences. If we assume moreover that H i ( X ; Q ) = 0 H_i(X;\mathbb Q)=0 , for all i ≫ 0 i\gg 0 , it is a classical result that the sequence of Betti numbers ( b i ) (b_i) grows polynomially or exponentially, depending on whether the sequence ( r i ) (r_i) is eventually zero or not. The purpose of this note is to prove, in both cases, that the r t h r^{\mathrm {th}} Betti number b r b_r is controlled by the immediately preceding ones. The proof of this result is based on a careful analysis of the Sullivan model of the free loop space X S 1 X^{S^1} .