In this paper we prove that, if
H
γ
(
X
)
{H^\gamma }(X)
denotes the
r
r
th deRham cohomology group of a connected manifold
X
X
and if the cup product
H
1
(
X
)
∧
R
H
1
(
X
)
→
H
2
(
X
)
{H^1}(X){ \wedge _R}{H^1}(X) \to {H^2}(X)
is not injective, then
π
1
(
X
)
{\pi _1}(X)
is not abelian. As a corollary, if
b
r
{b_r}
is the
r
r
th Betti number, then
1
2
b
1
(
b
1
−
1
)
>
b
2
\frac {1}{2}{b_1}({b_1} - 1) > {b_2}
implies
π
1
(
X
)
{\pi _1}(X)
being nonabelian.