A theorem on injectivity of the cup product

Abstract

We prove that if a space X has abelian or sufficiently abelian fundamental group, then the cup product H 1 ( X ) ∧ H 1 ( X ) → H 2 ( X ) {H^1}(X) \wedge {H^1}(X) \to {H^2}(X) is injective, giving an inequality between the associated Betti numbers. This generalises to a theorem of injectivity of the k-fold cup product on H n ( X ) {H^n}(X) , given that the kth order Whitehead product on π n ( X ) {\pi _n}(X) is trivial or torsion.