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We prove that if a space A'has abelian or sufficiently abelian fundamental group, then the cup product H1(X)f\H1(X)-* H2(X) is injective, giving an inequality between the associated Betti numbers. This generalises to a theorem of injectivity of the £-fold cup product on H"(X), given that the kth order Whitehead product on trn(X) is trivial or torsion. Let X be a topological space, and let Hn(X) denote its «th singular integral homology group. Let s/ denote the class of groups 7r such thatdoi:10.2307/2038752 fatcat:la7insywizf3heaane6uxsqw4i