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}
.