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

Abstract:

To any path connected topological space $X$, such that $\operatorname {rk} H_i(X) <\infty$ for all $i\geq 0$, are associated the following two sequences of integers: $b_i= \operatorname {rk} H_i(\Omega X)$ and $r_i= \operatorname {rk} \pi _{i+1}(X)$. If $X$ 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 $H_i(X;\mathbb Q)=0$, for all $i\gg 0$, it is a classical result that the sequence of Betti numbers $(b_i)$ grows polynomially or exponentially, depending on whether the sequence $(r_i)$ is eventually zero or not. The purpose of this note is to prove, in both cases, that the $r^{\mathrm {th}}$ Betti number $b_r$ 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 $X^{S^1}$.