A Sufficient Condition for Nonabelianness of Fundamental Groups of Differentiable Manifolds

In this paper we prove that, if <!-- MATH ${H^\gamma }(X)$ --> denotes the th deRham cohomology group of a connected manifold and if the cup product <!-- MATH ${H^1}(X){ \wedge _R}{H^1}(X) \to {H^2}(X)$ --> is not injective, then <!-- MATH ${\pi _1}(X)$ --> is not abelian. As a corollary, if is the th Betti number, then <!-- MATH $\frac{1}{2}{b_1}({b_1} - 1) > {b_2}$ --> {b_2}$"> implies <!-- MATH ${\pi _1}(X)$ --> being nonabelian.